There are many combinations of 2 students that can share their birthday In fact

# There are many combinations of 2 students that can

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There are many combinations of 2 students that can share their birthday. In fact, there are 22+21+20+…+1 combinations That is, 253 combinations Using the same logic, you can calculate that the probability of the event that 2 random Dutch citizens have a mutual acquaintance is pretty high.

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A primer on: binomial probabilities The number of ways of arranging k successes in a series of n observations (with constant probability p of success) is the number of possible combinations. This can be calculated with the binomial coefficient : In the birthday example: Where k = 0, 1, 2, ..., or n. )! ( ! ! k n k n n k n ! ( n k !) k ! 23! 21!2! 253
Binomial formulas The binomial coefficient “ n _choose_ k uses the factorial notation “ ! ”. The factorial n ! for any strictly positive whole number n is: n ! = n × ( n − 1) × ( n − 2) × · · · × 3 × 2 × 1 For example: 5 ! = 5 × 4 × 3 × 2 × 1 = 120 Note that 0! = 1.

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A Robbery in London Soho In London Soho, a woman is robbed. She and an eyewitness are able to describe the robber accurately. The robber is male, 2 metres tall, in between the age of 20 and 30, has red hair and has a pronounced limp. An eagle-eyed policeman notes later that day a man exactly fitting the description and arrests him. How likely is it that the policeman arrested the right person?
A Robbery in London Soho (2) When the man is brought to court the prosecutor hires a criminologist who reports the following figures: Being a male: P=0.51 Being 2 metres tall: P=0.025 Being in between 20 and 30y old: P=0.25 Being red-headed: P=0.037 Having a pronounced limp: P=0.017 Assuming independence, the multiplication rule tells us that the likelihood of a random person having these characteristics is 0.000002 (P(A)*P(B)*P(C) etc). Hence, the prosecutor argues that it is highly unlikely that the man is innocent.

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The Lawyer vs. the Statistician The arrested man decides not to hire a lawyer, but a statistician . The statistician argues that indeed the probability of a random person having these characteristics is 0.000002 However, since London has 10 million inhabitants, there must be 20 men fitting the description. Hence the likelihood of the man being the robber is not 99,999998%, but only 5%.
The prosecutor’s fallacy What mistake did the prosecutor make? The prosecutor mixed up 2 things: 1. the probability that the individual matches the description. 2. The probability that the individual who matches the description is also guilty. Clearly, the prosecutor needed a lecture in conditional probability.

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Conditional probability Multiplication rule for independent events: P(A and B) = P(A)*P(B) General multiplication rule for any 2 events: P(A and B) = P(A)*P(B|A) P(B|A) = conditional prob. that B occurs given that A has occured.
• Fall '17
• prosecutor, Monty Hall, Monty

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