3.38 Presentation (606)

Chirag Vithalani

February 21, 2016

  • 3.38 Male Children (Page 166) 
    
Male children. While it is often assumed that the probabilities of having a boy or a girl
are the same, the actual probability of having a boy is slightly higher at 0.51. Suppose a couple
plans to have 3 kids.

(a) Use the binomial model to calculate the probability that two of them will be boys.

(b) Write out all possible orderings of 3 children, 2 of whom are boys. Use these scenarios to
calculate the same probability from part (a) but using the addition rule for disjoint outcomes.
Confirm that your answers from parts (a) and (b) match.

(c) If we wanted to calculate the probability that a couple who plans to have 8 kids will have
3 boys, briefly describe why the approach from part (b) would be more tedious than the
approach from part (a).

  • The actual probability of have a boy is 0.51. Suppose a couple plans to have 3 kids.

    a) Use the binomial model to calculate the probability that two of them will be boys.

    • The probability of observing exactly k successes in n independent trials is given by - Binomial distribution -

      \(\frac{n!}{k!(n-k)!}p^k(1-p)^{n-k}\):

      probabilityOfBoy <- 0.51 #
k <- 2 #successes
n <- 3 #independent trials

factorialOfn <- factorial(n)
factorialOfk <- factorial(k)
factorialOfnminusk <- factorial(n-k)

probabilityOf2boysOf3 <- ( factorialOfn / (factorialOfk * factorialOfnminusk) ) * probabilityOfBoy^k * (1-probabilityOfBoy)^(n-k)
probabilityOf2boysOf3
      ## [1] 0.382347
      The probability that two of the three children will be boys is 0.3823.
    1. Write out all possible orderings of 3 children, 2 of whom are boys. Use these scenarios to calculate the same probability from part (a) by using the addition rule for disjoint outcomes. Confirm that your answers from parts (a) and (b) match.

    • Below table shows the possible ordering of 3 children with 2 boys-considering couple plans to have 3 kids:

      Child 1 | Child 2 | Child 3 |
--------|---------|---------|
Girl    | Boy     | Boy     |
Boy     | Girl    | Boy     |
Boy     | Boy     | Girl    |

      Now using the addition rule for disjoint outcomes : (i.e. since probability is same multiply with 3 )

      probabilityOf2boysOf3_B <- ((1-probabilityOfBoy) * probabilityOfBoy * probabilityOfBoy) * 3
probabilityOf2boysOf3_B
      ## [1] 0.382347

      As you can see, the result of (a) 0.3823 equals the result of (b) 0.3823.

    1. If we wanted to calculate the probability that a couple who plans to have 8 kids will have 3 boys, briefly describe why the approach from part (b) would be more tedious than the approach from part (a).

    • Short Answer : Because approach (a) uses formula.