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Fair Fairy Fare (Posted on 2009-12-10) Difficulty: 2 of 5
In a fairy tale, the fairy grandmother has assured the queen that her yet unborn baby prince will not die until the following set of conditions is fulfilled.

A scroll shall be prepared, and on the day of the birth and every subsequent birthday, a letter of the Greek alphabet from Α to Ω inclusively, chosen at random with replacement, shall be entered on the scroll. On the day that all the 24 distinct letters from Α to Ω of the alphabet appears, the prince will die.

Determine the expectation of the prince's life in years.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution Why 24 letters are a better alphabet (spoiler) | Comment 1 of 4
Well, in a bernoulli trial, where an event occurs with probability p, the expected number of repetitions until the event occurs is 1/p.  (I could derive this, but I won't).

Let p(n) be the probability of getting a new letter when there are already n distinct letters on the scroll.
p(n) = (24-n)/24.  P(1), for instance = 23/24

The expected length of time until getting a 2nd distinct letter = 1/p(1) = 24/23 years.
After that, the expected length of time until getting a 3rd distinct letter = 24/22 years.

Expected life of the prince = 24/23 + 24/22 + 24/21 + ... 24/2 + 24/1.

According to excel, this is approximately 89.623 years.

  Posted by Steve Herman on 2009-12-10 17:05:39
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