Probability question 2

Question

A person has two children. One is a boy born on a Tuesday. What is the probability that the other child is also a boy?

Answer

Let event A and event B be the births of the two children. Let us assume that the probability of a boy being born is equal to the probability of a girl being born (both 50%). The possible relevant outcomes are:

Event A – Boy born on a Tuesday (probability = (1/2)x(1/7) = 1/14)

Event A – Boy born on any day except Tuesday (probability = (1/2)x(6/7) = 6/14 = 3/7)

Event A – Girl born on any day (probability = 1/2)

and

Event B – Boy born on a Tuesday (probability = (1/2)x(1/7) = 1/14)

Event B – Boy born on any day except Tuesday (probability = (1/2)x(6/7) = 6/14 = 3/7)

Event B – Girl born on any day (probability = 1/2)

(Check: 1/2 + 3/7 + 1/14 = 1)

Independently of whether event A occurs before, after or simultaneously to event B, the only relevant combinations of these outcomes are:

  • Event A – Boy born on a Tuesday (probability = (1/2)x(1/7) = 1/14) AND Event B – Boy born on a Tuesday (probability = (1/2)x(1/7) = 1/14)
  • Event A – Boy born on a Tuesday (probability = (1/2)x(1/7) = 1/14) AND Event B – Boy born on any day except Tuesday (probability = (1/2)x(6/7) = 6/14 = 3/7)
  • Event A – Boy born on a Tuesday (probability = (1/2)x(1/7) = 1/14) AND Event B – Girl born on any day (probability = 1/2)
  • Event A – Boy born on any day except Tuesday (probability = (1/2)x(6/7) = 6/14 = 3/7) AND Event B – Boy born on a Tuesday (probability = (1/2)x(1/7) = 1/14)
  • Event A – Girl born on any day (probability = 1/2) AND Event B – Boy born on a Tuesday (probability = (1/2)x(1/7) = 1/14

 

The probabilities of each of these combinations are:

Probability of combination 1) = (1/14)(1/14) = 1/196

Probability of combination 2) = (1/14)(3/7) = 3/98

Probability of combination 3) = (1/14)(1/2) = 1/28

Probability of combination 4) = (3/7)(1/14) = 3/98

Probability of combination 5) = (1/2)(1/14) = 1/28

Combinations 1), 2) and 4) are formed by two boys, hence the probability of the second child being a boy is:

((1/196)+(3/98)+(3/98)) / ((1/196)+(3/98)+(1/28)+(3/98)+(1/28)) = (13/196) / (27/196) = 13/27

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