Please answer all questions in full details! Will Rate! 1 Finding Antony Smith Y

Question

Please answer all questions in full details! Will Rate!

1 Finding Antony Smith You are trying to locate an old high school friend who lives in Chicago. Unfortunately, your friend's name is Anthony Smith and the Chicago phone book lists phone numbers for 24 different people named Anthony Amith a) If you call 10 of these Anthony Smith's at random, what is the probability that you will call your friend? (Assume that your friend's phone number is listed in the phone book, and that you don't call b) Let X be the number of calls you need to make until you find your friend. Give the probability mass c) How many calls do you expect to make until you find your friend? (Again, assume that your friend's anybody twice.) function for X. phone number is listed in the phone book, and that you don't call anybody twice.)

Explanation / Answer

a) There are 24 different Anthony Smiths.

The probability that the first call will ring the right Anthony Smith = 1/24

The probability that the second call will ring the right Anthony Smith = 1/24

........

The probability that the 10th call will ring the right Anthony Smith = 1/24

=> Probability that one of these 10 calls will ring the right Anthony Smith = 1/24 * 10 = 10/24

= 0.417

b) We have P (X = 1) = 1/24

P (X = 2) = 2/24

.....

P (X = 24) = 24/24

The probability mass function is P(X) = x/24.

c) Probability that one call will ring the friend = 1/24

Probability that two calls will ring the friend = 2/24

......

Probability that 24 calls will ring the friend = 24/24

=> Expected number of calls = 1/24 + 2/24 + 3/24 + .... 24/24

= 1/24 (1 + 2 + 3 + .... 24)

By formula for sum of squares = n (n+1) / 2

= 1/24 * 24 * 25 / 2

= 12.5

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