2014 Crime in the United States report

According to the 2014 Crime in the United States report, while most blacks are indeed killed by other blacks (90%), whites are mostly killed by whites (82.4%). Further, only 14% of whites in 2014 were killed by blacks and 7.6% … Continue reading

According to the 2014 Crime in the United States report, while most blacks are indeed killed by other blacks (90%), whites are mostly killed by whites (82.4%). Further, only 14% of whites in 2014 were killed by blacks and 7.6% of blacks in 2014 were killed by whites.


religion and terrorism

Muslim Opinion Polls A “Tiny Minority of Extremists”? “Strive hard against the unbelievers and the hypocrites and be unyielding to them; and their abode is hell, and evil is their destination.” Quran 9:73   May 03 2013 Killing Civilians Is More … Continue reading

Muslim Opinion Polls

A “Tiny Minority of Extremists”?

“Strive hard against the unbelievers and the hypocrites and be
unyielding to them; and their abode is hell, and evil is their destination.”

Quran 9:73

 


May 03 2013

Killing Civilians Is More Popular Than You’d Think–Especially Among Pundits

I came across this polling from Gallup (8/2/11) while I was looking to debunk the nutty idea that Muslim Americans never criticize terrorism.

As the Gallup poll shows, of all religious groups surveyed–including nonbelievers–Muslims are the least likely to say it’s OK to kill civilians


America’s Breivik Complex: State terror and the Islamophobic right

Few political terrorists in recent history took as much care to articulate their ideological influences and political views as Anders Behring Breivik did. The right-wing Norwegian Islamophobe who murdered 76 children and adults in Oslo and at a government-run youth camp spent months, if not years, preparing his 1,500 page manifesto.

Besides its length, one of the most remarkable aspects of the manifesto is the extent to which its European author quoted from the writings of figures from the American conservative movement. Though he referred heavily to his fellow Norwegian, the blogger Fjordman, it was Robert Spencer, the American Islamophobic pseudo-academic, who received the most references from Breivik — 55 in all. Then there was Daniel Pipes, the Muslim-bashing American neoconservative who earned 18 citations from the terrorist. Other American anti-Muslim characters appear prominently in the manifesto, including the extremist blogger Pam Geller, who operates an Islamophobic organization in partnership with Spencer.

The modern Islamic fundamentalist movements have their origins in the late 19th century.[30] The Wahhabi movement, an Arabian fundamentalist movement that began in the 18th century, gained traction and spread during the 19th and 20th centuries.[31] During the Cold War following World War II, some NATO governments, particularly those of the United States and the United Kingdom, launched covert and overt campaigns to encourage and strengthen fundamentalist groups in the Middle East and southern Asia. These groups were seen as a hedge against potential expansion by the Soviet Union, and as a means to prevent the growth of nationalistic movements that were not necessarily favorable toward the interests of the Western nations.[32] By the 1970s the Islamists had become important allies in supporting governments, such as Egypt, which were friendly to U.S. interests. By the late 1970s, however, some fundamentalist groups had become militaristic leading to threats and changes to existing regimes. The overthrow of the Shah in Iran and rise of the Ayatollah Khomeini was one of the most significant signs of this shift.[33] Subsequently fundamentalist forces in Algeria caused a civil war, caused a near-civil war in Egypt, and caused the downfall of the Soviet occupation in Afghanistan.[34] In many cases the military wings of these groups were supplied with money and arms by the U.S. and U.K.

Published on Dec 10, 2015

By the Numbers is an honest and open discussion about Muslim opinions and demographics. Narrated by Raheel Raza, president of Muslims Facing Tomorrow, this short film is about the acceptance that radical Islam is a bigger problem than most politically correct governments and individuals are ready to admit. Is ISIS, the Islamic State, trying to penetrate the U.S. with the refugee influx? Are Muslims radicalized on U.S. soil? Are organizations such as CAIR, who purport to represent American Muslims accepting and liberal or radicalized with links to terror organizations?

It’s time to have your say, go to http://go.clarionproject.org/numbers-…


OUT OF PROPORTION

01.14.155:45 AM ET
“Not all Muslims are terrorists, but all terrorists are Muslims.” How many times have you heard that one? Sure, we heard Fox News’s Brian Kilmeade say it.
Want to guess what percent of the terrorist attacks there were committed by Muslims over the past five years? Wrong. That is, unless you said less than 2 percent.
Per the 2013 State Department’s report on terrorism, there were 399 acts of terror committed by Israeli settlers in what are known as “price tag” attacks. These Jewish terrorists attacked Palestinian civilians causing physical injuries to 93 of them and also vandalized scores of mosques and Christian churches.

Back in the United States, the percentage of terror attacks committed by Muslims is almost as miniscule as in Europe. An FBI study looking at terrorism committed on U.S. soil between 1980 and 2005 found that 94 percent of the terror attacks were committed by non-Muslims. In actuality, 42 percent of terror attacks were carried out by Latino-related groups, followed by 24 percent perpetrated by extreme left-wing actors.

And as a 2014 study by University of North Carolina found, since the 9/11 attacks, Muslim-linked terrorism has claimed the lives of 37 Americans. In that same time period, more than 190,000 Americans were murdered (PDF).


Hill’s criteria for causation

The Bradford Hill criteria, otherwise known as Hill’s criteria for causation, are a group of minimal conditions necessary to provide adequate evidence of a causal relationship between an incidence and a possible consequence, established by the English epidemiologist Sir Austin Bradford Hill (1897–1991) in 1965. The list of the criteria is as follows: Strength (effect […]

The Bradford Hill criteria, otherwise known as Hill’s criteria for causation, are a group of minimal conditions necessary to provide adequate evidence of a causal relationship between an incidence and a possible consequence, established by the English epidemiologist Sir Austin Bradford Hill (1897–1991) in 1965.

The list of the criteria is as follows:

  1. Strength (effect size): A small association does not mean that there is not a causal effect, though the larger the association, the more likely that it is causal.[1]
  2. Consistency (reproducibility): Consistent findings observed by different persons in different places with different samples strengthens the likelihood of an effect.[1]
  3. Specificity: Causation is likely if there is a very specific population at a specific site and disease with no other likely explanation. The more specific an association between a factor and an effect is, the bigger the probability of a causal relationship.[1]
  4. Temporality: The effect has to occur after the cause (and if there is an expected delay between the cause and expected effect, then the effect must occur after that delay).[1]
  5. Biological gradient: Greater exposure should generally lead to greater incidence of the effect. However, in some cases, the mere presence of the factor can trigger the effect. In other cases, an inverse proportion is observed: greater exposure leads to lower incidence.[1]
  6. Plausibility: A plausible mechanism between cause and effect is helpful (but Hill noted that knowledge of the mechanism is limited by current knowledge).[1]
  7. Coherence: Coherence between epidemiological and laboratory findings increases the likelihood of an effect. However, Hill noted that “… lack of such [laboratory] evidence cannot nullify the epidemiological effect on associations”.[1]
  8. Experiment: “Occasionally it is possible to appeal to experimental evidence”.[1]
  9. Analogy: The effect of similar factors may be considered.[1]

Debate in modern epidemiology

Bradford Hill’s criteria are still widely accepted in the modern era as a logical structure for investigating and defining causality in epidemiological study. However, their method of application is debated. Some proposed options include:

  1. using a counterfactual consideration as the basis for applying each criterion.[2]
  2. subdividing them into three categories: direct, mechanistic and parallel evidence, expected to complement each other. This operational reformulation of the criteria has been recently proposed in the context of evidence based medicine.[3]
  3. considering confounding factors and bias.[4]
  4. using Hill’s criteria as a guide but not considering them to give definitive conclusions.[5]
  5. separating causal association and interventions, because interventions in public health are more complex than can be evaluated by use of Hill’s criteria[6]

Arguments against the use of Bradford Hill criteria as exclusive considerations in proving causality also exist. Some argue that the basic mechanism of proving causality is not in applying specific criteria—whether those of Bradford Hill or counterfactual argument—but in scientific common sense deduction.[7] Others also argue that the specific study from which data has been produced is important, and while the Bradford-Hill criteria may be applied to test causality in these scenarios, the study type may rule out deducing or inducing causality, and the criteria are only of use in inferring the best explanation of this data.[8]

Debate over the scope of application of the criteria includes whether they can be applied to social sciences.[9] The argument proposed in this line of thought is that when considering the motives behind defining causality, the Bradford Hill criteria are important to apply to complex systems such as health sciences because they are useful in prediction models where a consequence is sought; explanation models as to why causation occurred are deduced less easily from Bradford Hill criteria as the instigation of causation, rather than the consequence, is needed for these models.

Researchers have applied Hill’s criteria for causality in examining the evidence in several areas of epidemiology, including connections between ultraviolet B radiation, vitamin D and cancer,[10][11] vitamin D and pregnancy and neonatal outcomes,[12] alcohol and cardiovascular disease outcomes,[13] infections and risk of stroke,[14] nutrition and biomarkers related to disease outcomes,[15] and sugar-sweetened beverage consumption and the prevalence of obesity and obesity-related diseases.[16] Referenced papers can be read to see how Hill’s criteria have been applied.


dplyr

dplyr is a new package which provides a set of tools for efficiently manipulating datasets in R. dplyr is the next iteration of plyr, focussing on only data frames. dplyr is faster, has a more consistent API and should be easier to use. There are three key ideas that underlie dplyr: Your time is important, … Continue reading dplyr

dplyr is a new package which provides a set of tools for efficiently manipulating datasets in R. dplyr is the next iteration of plyr, focussing on only data frames. dplyr is faster, has a more consistent API and should be easier to use. There are three key ideas that underlie dplyr:

  1. Your time is important, so Romain Francois has written the key pieces in Rcpp to provide blazing fast performance. Performance will only get better over time, especially once we figure out the best way to make the most of multiple processors.
  2. Tabular data is tabular data regardless of where it lives, so you should use the same functions to work with it. With dplyr, anything you can do to a local data frame you can also do to a remote database table. PostgreSQL, MySQL, SQLite and Google bigquery support is built-in; adding a new backend is a matter of implementing a handful of S3 methods.
  3. The bottleneck in most data analyses is the time it takes for you to figure out what to do with your data, and dplyr makes this easier by having individual functions that correspond to the most common operations (group_by, summarise, mutate, filter, select and arrange). Each function does one only thing, but does it well.

OpenBUGS

BUGS is a software package for performing Bayesian inference Using Gibbs Sampling. The user specifies a statistical model, of (almost) arbitrary complexity, by simply stating the relationships between related variables. The software includes an ‘expert system’, which determines an appropriate MCMC (Markov chain Monte Carlo) scheme (based on the Gibbs sampler) for analysing the specified … Continue reading OpenBUGS

BUGS is a software package for performing Bayesian inference Using Gibbs Sampling. The user specifies a statistical model, of (almost) arbitrary complexity, by simply stating the relationships between related variables. The software includes an ‘expert system’, which determines an appropriate MCMC (Markov chain Monte Carlo) scheme (based on the Gibbs sampler) for analysing the specified model. The user then controls the execution of the scheme and is free to choose from a wide range of output types.

Versions…

There are two main versions of BUGS, namely WinBUGS and OpenBUGS. This site is dedicated to OpenBUGS, an open-source version of the package, on which all future development work will be focused. OpenBUGS, therefore, represents the future of the BUGS project. WinBUGS, on the other hand, is an established and stable, stand-alone version of the software, which will remain available but not further developed. The latest versions of OpenBUGS (from v3.0.7 onwards) have been designed to be at least as efficient and reliable as WinBUGS over a wide range of test applications. Please see here for more information on WinBUGS. OpenBUGS runs on x86 machines with MS Windows, Unix/Linux or Macintosh (using Wine).

Note that software exists to run OpenBUGS (and analyse its output) from within both R and SAS, amongst others.

For additional details on the differences between OpenBUGS and WinBUGS see the OpenVsWin manual page.

to read Excel files from R

Many solutions have been implemented to read Excel files from R: each one has advantages and disadvantages, so an universal solution is not available. Get an overview of all the solutions, allows the choice of the best solution case-by-case.

Many solutions have been implemented to read Excel files from R: each one has advantages and disadvantages, so an universal solution is not available. Get an overview of all the solutions, allows the choice of the best solution case-by-case.

reshape2

Sean C. Anderson
October 19, 2013
An Introduction to reshape2
reshape2 is an R package written by Hadley Wickham that makes it easy to transform data between wide and long formats.

Sean C. Anderson

October 19, 2013
An Introduction to reshape2
reshape2 is an R package written by Hadley Wickham that makes it easy to transform data between wide and long formats.

ggplot2

ggplot2 is a data visualization package for the statistical programming language R. Created by Hadley Wickham in 2005, ggplot2 is an implementation of Leland Wilkinson‘s Grammar of Graphics—a general scheme for data visualization which breaks up graphs into semantic components such as scales and layers. ggplot2 can serve as a replacement for the base graphics in … Continue reading ggplot2

ggplot2 is a data visualization package for the statistical programming language R. Created by Hadley Wickham in 2005, ggplot2 is an implementation of Leland Wilkinson‘s Grammar of Graphics—a general scheme for data visualization which breaks up graphs into semantic components such as scales and layers. ggplot2 can serve as a replacement for the base graphics in R and contains a number of defaults for web and print display of common scales. Since 2005, ggplot2 has grown in use to become one of the most popular R packages.[1][2] It is licensed under GNU GPL v2.[3]

On 2 March 2012, ggplot2 version 0.9.0 was released with numerous changes to internal organization, scale construction and layers.[4] An update dealing primarily with bug fixes was released on 9 May 2012, incrementing the version to 0.9.1.[5]

On 25 February 2014, Hadley Wickham formally announced that “ggplot2 is shifting to maintenance mode. This means that we are no longer adding new features, but we will continue to fix major bugs, and consider new features submitted as pull requests. In recognition this significant milestone, the next version of ggplot2 will be 1.0.0″.[6]

Monty Hall problem

Bertrand’s box paradox is a classic paradox of elementary probability theory. It was first posed by Joseph Bertrand in his Calcul des probabilités, published in 1889. There are three boxes: a box containing two gold coins, a box containing two silver coins, a box containing one gold coin and one silver coin. After choosing a box at random and withdrawing one coin […]

Bertrand’s box paradox is a classic paradox of elementary probability theory. It was first posed by Joseph Bertrand in his Calcul des probabilités, published in 1889.

There are three boxes:

  1. a box containing two gold coins,
  2. a box containing two silver coins,
  3. a box containing one gold coin and one silver coin.

After choosing a box at random and withdrawing one coin at random, if that happens to be a gold coin, it may seem that the probability that the remaining coin is gold is 1?2; in fact, the probability is actually 2?3. Two problems that are very similar are the Monty Hall problem and the Three Prisoners problem.

These simple but slightly counterintuitive puzzles are used as a standard example in teaching probability theory. Their solution illustrates some basic principles, including theKolmogorov axioms.


The Monty Hall problem is a brain teaser, in the form of a probability puzzle (Gruber, Krauss and others), loosely based on the American television game show Let’s Make a Deal and named after its original host, Monty Hall. The problem was originally posed in a letter by Steve Selvin to the American Statistician in 1975 (Selvin 1975a), (Selvin 1975b). It became famous as a question from a reader’s letter quoted inMarilyn vos Savant‘s “Ask Marilyn” column in Parade magazine in 1990 (vos Savant 1990a):

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Vos Savant’s response was that the contestant should switch to the other door. (vos Savant 1990a)

The argument relies on assumptions, explicit in extended solution descriptions given by Selvin (1975b) and by vos Savant (1991a), that the host always opens a different door from the door chosen by the player and always reveals a goat by this action—because he knows where the car is hidden. Leonard Mlodinow stated: “The Monty Hall problem is hard to grasp, because unless you think about it carefully, the role of the host goes unappreciated.” (Mlodinow 2008) It is also assumed that the contestant prefers to win a car, rather than a goat.

Contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their choice have only a 1/3 chance. One explanation notices that 2/3 of the time, the initial choice of the player is a door hiding a goat. The host is then forced to open the other goat door, and the remaining one must, therefore, hide the car. “Switching” only fails to give the car when the player picks the “right” door to begin with, which only has a 1/3 chance.

Many readers of vos Savant’s column refused to believe switching is beneficial despite her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them claiming vos Savant was wrong (Tierney 1991). Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy (vos Savant 1991a). Paul Erd?s, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation confirming the predicted result (Vazsonyi 1999).

The Monty Hall problem has attracted academic interest from the surprising result and simple formulation. Variations of the Monty Hall problem are made by changing the implied assumptions and can create drastically different consequences. For one variation, if Monty only offers the contestant a chance to switch when the contestant initially chose the door hiding the car, then the contestant should never switch. For another variation, if Monty opens another door randomly and happens to reveal a goat, then it makes no difference (Rosenthal, 2005a), (Rosenthal, 2005b).

The problem is a paradox of the veridical type, because the correct result (you should switch doors) is so counterintuitive it can seem absurd, but is nevertheless demonstrably true. The Monty Hall problem is mathematically closely related to the earlier Three Prisoners problem and to the much older Bertrand’s box paradox.

The problem continues to attract the attention of cognitive psychologists. The typical behaviour of the majority, i.e., not switching, may be explained by phenomena known in the psychological literature as: 1) the endowment effect (Kahneman et al., 1991); people tend to overvalue the winning probability of the already chosen – already “owned” – door; 2) the status quo bias (Samuelson and Zeckhauser, 1988); people prefer to stick with the choice of door they have already made; 3) the errors of omission vs. errors of commission effect (Gilovich et al., 1995); all else considered equal, people prefer that any errors that they are responsible for to have occurred through ‘omission’ of taking action rather than through having taken an explicit action that later becomes known to have been erroneous. Experimental evidence confirms that these are plausible explanations which do not depend on probability intuition (Kaivanto et al., 2014Morone and Fiore, 2007).

Criticism of the simple solutions

As already remarked, most sources in the field of probability, including many introductory probability textbooks, solve the problem by showing the conditional probabilities the car is behind door 1 and door 2 are 1/3 and 2/3 (not 1/2 and 1/2) given the contestant initially picks door 1 and the host opens door 3; various ways to derive and understand this result were given in the previous subsections. Among these sources are several that explicitly criticize the popularly presented “simple” solutions, saying these solutions are “correct but … shaky” (Rosenthal 2005a), or do not “address the problem posed” (Gillman 1992), or are “incomplete” (Lucas et al. 2009), or are “unconvincing and misleading” (Eisenhauer 2001) or are (most bluntly) “false” (Morgan et al. 1991). Some say that these solutions answer a slightly different question – one phrasing is “you have to announce before a door has been opened whether you plan to switch” (Gillman 1992, emphasis in the original).

The simple solutions show in various ways that a contestant who is determined to switch will win the car with probability 2/3, and hence that switching is the winning strategy, if the player has to choose in advance between “always switching”, and “always staying”. However, the probability of winning by always switching is a logically distinct concept from the probability of winning by switching given the player has picked door 1 and the host has opened door 3. As one source says, “the distinction between [these questions] seems to confound many” (Morgan et al. 1991). This fact that these are different can be shown by varying the problem so that these two probabilities have different numeric values. For example, assume the contestant knows that Monty does not pick the second door randomly among all legal alternatives but instead, when given an opportunity to pick between two losing doors, Monty will open the one on the right. In this situation the following two questions have different answers:

  1. What is the probability of winning the car by always switching?
  2. What is the probability of winning the car given the player has picked door 1 and the host has opened door 3?

The answer to the first question is 2/3, as is correctly shown by the “simple” solutions. But the answer to the second question is now different: the conditional probability the car is behind door 1 or door 2 given the host has opened door 3 (the door on the right) is 1/2. This is because Monty’s preference for rightmost doors means he opens door 3 if the car is behind door 1 (which it is originally with probability 1/3) or if the car is behind door 2 (also originally with probability 1/3). For this variation, the two questions yield different answers. However as long as the initial probability the car is behind each door is 1/3, it is never to the contestant’s disadvantage to switch, as the conditional probability of winning by switching is always at least 1/2. (Morgan et al. 1991)


Players who STAY have won 49040 cars out of 145987 games yielding a winning percentage of 34%
players who SWITCH have won 68356 cars out of 103063 games yielding a winning percentage of 66%

Video Transcript:

You’re on a game show and there are three doors in front of you. The host, Monty Hall, says, “Behind one door is a brand new car. Behind the other two doors are goats. Pick a door!” You think, “Well, it doesn’t matter which door I choose, every door has a 1/3 chance of having the car behind it.” So, you choose door number 1. Now it gets interesting. Monty, the host, who knows where the car is, opens door number 2 and reveals a goat. The host always opens a door to reveal a goat. The host says, “If you want, you can switch to door number 3.” What should you do? Stay with your original choice or switch to the other door? All right, so what are you going to do? Stay or switch? Well, it’s a fifty-fifty chance of winning the car in either door. Right? [Wrong!] You actually double your chances of winning the car by switching doors. And that is why the Monty Hall Problem is so evasive!
Choose an explanation to the Monty Hall Problem:

1/3 vs 2/3 – Solution #1 to the Monty Hall Problem
There is a 1/3 chance of the car being behind door number 1 and a 2/3 chance that the car isn’t behind door number 1. After Monty Hall opens door number 2 to reveal a goat, there’s still a 1/3 chance that the car is behind door number 1 and a 2/3 chance that the car isn’t behind door number 1. A 2/3 chance that the car isn’t behind door number 1 is a 2/3 chance that the car is behind door number 3.
100 Doors! – Solution #2 to the Monty Hall Problem
Imagine that instead of 3 doors, there are 100. All of them have goats except one, which has the car. You choose a door, say, door number 23. At this point, Monty Hall opens all of the other doors except one and gives you the offer to switch to the other door. Would you switch? Now you may arrogantly think, “Well, maybe I actually picked the correct door on my first guess.” But what’s the probability that that happened? 1/100. There’s a 99% chance that the car isn’t behind the door that you picked. And if it’s not behind the door that you picked, it must be behind the last door that Monty left for you. In other words, Monty has helped you by leaving one door for you to switch to, that has a 99% chance of having the car behind it. So in this case, if you were to switch, you would have a 99% chance of winning the car.
Pick a Goat – Solution #3 to the Monty Hall Problem
To win using the stay strategy, you need to choose the car on your first pick because you’re planning to stay with your initial choice. The chance of picking the car on your first pick is clearly one out of three. But, in order to win using the switch strategy, you only need to pick a goat on your first pick because the host will reveal the other goat and you’ll end up switching to the car. So you want to use the strategy that lets you win if you choose a goat initially because you’re twice as likely to start by picking a goat.
Scenarios – Solution #4 to the Monty Hall Problem
To understand why it’s better to switch doors, let’s play out a few scenarios. Let’s see what will happen if you were to always stay with your original choice. We’ll play out three scenarios, one for each door that the car could be behind (door number 1, door number 2, or door number 3). And it doesn’t matter which door you start out with, so, to keep it simple, we’ll always start by choosing door number 1.
Stay strategy, scenario 1: the car is behind door number 1. You choose door number 1, then the host reveals a goat behind door number 2 and because you always stay, you stay with door number 1. You win the car! Stay strategy, scenario 2: the car is behind door number 2. You start by picking door number 1, the host reveals a goat behind door number 3, and you’re using the stay strategy so you stay with door number 1. You get a goat and don’t win the car. Stay strategy, scenario 3: the car is behind door number 3. You pick door number 1, the host opens door number 2 to reveal a goat, you stay with door number 1, and you get a goat. So, using the stay strategy, you won the car one out of three times. That means that in any one instance of playing the game, your chance of winning the car if you choose to stay is 1/3 or about 33%.

Now let’s try switching doors. Again, we’ll always start by picking door number 1. Switch strategy, scenario 1: the car is behind door number 1. You choose door number 1, the host opens door number 2 to reveal a goat, you are using the switch strategy so you switch to door number 3. You get a goat. Switch strategy, scenario 2: the car is behind door number 2. You start by picking door number 1, the host opens door number 3 to reveal a goat, you switch to door number 2 and win the car! Switch strategy, scenario 3: the car is behind door number 3. You pick door number 1, the host opens door number 2 to reveal a goat, you switch to door number 3 and win the car again! So, with the switch strategy you won the car 2 out of 3 times. That means, that in any one instance of the game, your chance of winning the car if you choose to switch doors is 2/3 or about 67%.

Therefore, if you play the game three times and stay, on average you’ll win the car once. But if you play the game three times and switch each time, on average you’ll win the car twice. That’s twice as many cars!