Example 219 the monty hall problem the problem is

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Example 2.19 (The Monty Hall problem). The problem is named for the host of the television show Let’s Make A Deal in which contestants were often placed in situations like the following: Three curtains are numbered 1, 2, and 3. Behind one curtain is a car; behind the other two curtains are donkeys. You
pick a curtain, say #1. To build some suspense the host opens up one of the two remaining curtains, say #3, to reveal a donkey. What is the probability you will win given that there is a donkey behind #3? Should you switch curtains and pick #2 if you are given the chance?
2.2. CONDITIONAL PROBABILITY 47 Although it took a number of steps to compute this answer, it is “obvious.” When we picked one of the three doors initially we had probability 1/3 of picking the car, and since the host can always open a door with a donkey the new information does not change our chance of winning. The paradox actually predates the game show in the following form. Three prisoners, Al, Bob, and Charlie, are in a cell. At dawn two will be set free and one will be hanged, but they do not know who will be chosen. The guard offers to tell Al the name of one of the other two prisoners who will go free but Al stops him, screaming, “No, don’t! That would increase my chances of being hanged to 1/2.” Criticize Al’s reasoning. People vs. Collins. In 1964 the purse of an elderly woman shopping in Los Angeles was snatched by a young white female with a blond ponytail. The theif fled on foot but soon after was seen getting into a yellow car driven by a black man who had a mustache and beard. A police investigation subsequently turned up a suspect who was blond, wore a ponytail, and lived with a black man who had a mustache, beard, and a yellow car. None of the eyewitnesses were able to identify the suspects so the police turned to probability Characteristic Probability Yellow car 0.1 Man with mustache 0.25 Black man with beard 0.01 Woman with ponytail 0.1 Woman with blond hair 0.33 Interracial couple in car 0.01 Multiplying the probabilities as if they were independent gives an overall prob- ability of 1 / 12 , 000 , 000. This impressed the jury who handed down a verdict of second degree robbery. The Supreme Court of California later disagreed and

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