A royal family has children until it has a boy or until it has three children. Assume that each child is a boy with probability 1/2. Find the expected number of boys in this royal family and the expected number of girls.
With 3 children the probabilities are:
1 boy and 2 girls, or 1 girl and 2 boys =37.50%=N!/(B!.G! x 2^N) x 100
2 boys and 1 girl or 2 girls and 1 boy =37.50% =,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
3 boys and no girls or 3 girls and no boys=12.50%=,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
+33653 p(0g) = 1/2. (Probability of no girls)
p(1g) = 1/2^2 (Probability of exactly 1 girl)
p(2g) = 1/2^3. (Probability of exactly 2 girls)
p(3g) = 1/2^3. (Probability of exactly 3 girls)
Expected number of girls = 0*1/2 + 1*1/2^2 + 2*1/2^3 + 3*1/2^3 → 7/8
p(0b) = 1/2^3
p(1b) = 1/2^3 + 1/2^2 + 1/2 → 7/8
Expected number of boys = 0*1/2^3 + 1*7/8 → 7/8
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