The probability that a child being born will be a girl is 50%. Suppose couple re

Question

The probability that a child being born will be a girl is 50%. Suppose couple really wants to have a girl.

(a) What is the expected number of children until they get a daughter?

(b) What is the probability that their first daughter will be the fourth child?

(c) What is the probability that you will have to have at least six children but fewer than ten in order to get their first daughter?

(d) Suppose that the couple already had four sons. What is the probability that their first daughter will be their seventh child?

Suppose you are dealt five cards from a well shuffled standard 52-card deck of cards.

(a) What is the probability that you will get exactly four diamonds?

(b) What is the probability that you will be dealt at least three Aces?

Explanation / Answer

a) expected number of children is 1/0.50 = 2
b) The probability that their first daughter will be the fourth child is
= P(first 3 childerns are boys) P(4th child is girl)
= (0.5*0.5*0.5)*(0.5) = 0.0625
c) P(6<= X < 10) = P(X=6) + P(X=7) + P(X=8) + P(X=9)
= 0.5^6 + 0.5^7 + 0.5^8 +0.5^9 = 0.0293
d) P(X=7 / already had four sons) = P(X=3) lack of memory less property
= 0.5^3 = 0.125

a) Probability that you will get exactly four diamonds is 13C4 * 39C1 / 52C5 = 0.01073

b) P(X>=3) = P(X=3) + P(X=4) + P(X=5)
= (4C3 * 48C2) / 52C5 + (4C4 * 48C1) / 52C5 + 0
= ( 4512 + 48 ) / 2598960 = 0.0017545  

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