Disney Magic

Probability Level 4

I was at Disneyland and decided to conduct a survey of the families. I asked people at random if they have exactly two children. And if so, if one of them was a boy born in January.

After a while, I finally found someone who said yes. What is the probability that this person has two boys?

Assume that there is an equal chance of either sex, and equal chance of being born in any month of the year.

\frac{13}{27}

\frac{1}{2}

\frac{23}{47}

\frac{3}{7}

1 solution

Pranshu Gaba
Aug 11, 2015

Our sample space is all families with two children, one of which is a boy born in January. In each family, there will be an older sibling and a younger sibling. We will now list out all the cases in the sample space and select the favorable ones to find out the probability. I will name the older sibling A and the younger sibling B.

We can have the following cases:

Case 1: A and B are both boys, both are born in January.
Case 2: A and B are both boys, A is born in January and B is born in any of the other 11 months.
Case 3: A and B are both boys, B is born in January, and A is born in any of the other 11 months.
Case 4: A is a boy born in January, B is a girl born in any of the 12 months.
Case 5: B is a boy born in January, A is a girl born in any of the 12 months.

Note that Cases 1, 2, 3 are favorable.

The relative probability of each case occurring is in the ratio $1 : 11 : 11 : 12 : 12$ .

Therefore the probability of both children being boys is $\dfrac{1 + 11 + 11}{1 + 11 +11 + 12 +12 }$ $= \boxed{\dfrac{23 }{47}}$ $~~_\square$

Moderator note:

Good approach with understanding the sample space and using that to help us understand this conditional probability question.

Disney Magic

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