A royal family has children until it has a boy or until it has three children, whichever comes first. 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 boys.

• 0.825
• 0.85
• 0.875
• 0.9
• 0.925

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: C

The number of boys say [math]N[/math] is less than equal to 1. The probability that there are no boys only occurs when we have 3 girls and this equals (1/2)³. Hence the expected number of boys equals 1-0.125 = 0.875.