(Intuition) In many scientific experiments, a result inferred from observation w
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Question
(Intuition) In many scientific experiments, a result inferred from observation with 95% confidence (i.e., 95% likelihood of being true) is often consider a 'sure thing'. In the birthday problem (discussed at our first lecture), the probability of there being at least one matching birthday among n people was shown to be Pr{at least one matching birthday among n people} = 1 - (365)(364)(363)...(365 - n + 1)/365^n for positive integer n. We discussed in lecture how we only need 23 people to make this probability approximately 50%: in fact, we only need 47 people to make it a 'sure thing' (i.e., make the probability of at least one match around 95%)!! Yet with 365 days in a common year, our intuition tells us it should be a reasonably possible event that all 47 people would have different birthdays: however, the probability formula tells us differently. Your assignment for this problem is to provide some insight as to why our intuition fails: that is, explain why it is a near certainty that if 47 people attend a party at least two of them share a birthday. Consider both the context of the birthday problem itself as well as (1) above.Explanation / Answer
The intuition, and in turn experience, is based on the personal experience. Now, the birthday problem requires that the people chosen are completely random. However, mostly in schools and colleges, we don't find "our" birthday mate in spite of the population exceeding 50. Note that the classroom strength is not necessarily a collection of truly independent events as the admission cut off is determined by a specific month of the year.
On the other hand, (1) is an answer to finding a birthday mates and not "your birthday mate". That is the reason for the intuition failing.
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