Conditional Probability Boy Girl

the other child is a girl is correct. 

Suppose we're told that the oldest child is a boy? Let's look at this new information using the same two  methods:  Combinations in the Sample Space  When the oldest child is a boy, the original sample space:  {BB, BG, GB, GG}  shrinks to:  {BB, BG}  because the two other possibilities, {GB, GG}, show girls as the oldest child and thus are no  longer possible.  Since only one of the possibilities in the new sample space, {BG}, includes a girl, the probability that the second child in the family is a girl is 1/2. Probability Tree  For this variation of the problem we construct the same conditional probability tree as above, but  since in our notation the first letter represents the oldest child, we delete the bottom two paths  where both combinations represent an oldest daughter:  First

Second

Child

Child

__B(1/2)__

Unconditional Conditional Probability

Probability

1/4

1/2

1/4

1/2

| __B(1/2)__| |

|__G(1/2)__

| Family -| 

To find the probability that the younger child is a girl, follow the second path. The probability is  1/2.  Remember: information that creates conditional probability can dramatically affect common sense ideas  about probability. For example, no matter how unlikely it may seem to you, if you meet a mother of two  who says she has a daughter, a basic knowledge of probability tells you there's a 2/3 probability that the  daughter mentioned has a brother. If she says she's an older daughter, you know there's a 1/2 probability  that the daughter has a younger brother.

boys and girls are equally likely to occur. Here's an unconditional probability tree for a two-child family: First

Second

Child

Child

__B(1/2)__

Unconditional Probability

1/4

| __B(1/2)__| |

|__G(1/2)__

1/4

__B(1/2)__

1/4

| Family -| |

|__G(1/2)__| | |__G(1/2)__

1/4 

To turn an unconditional probability tree into a conditional probability tree, we delete the paths  that are not relevant to the new conditions given with the problem. The unconditional probability  for each combination remains the same, but the conditional probability is calculated from the new  revised sample space.  First

Second

Child

Child

__B(1/2)__

Unconditional Probability

Conditional

Probability

1/4

1/3

|__G(1/2)__

1/4

1/3

__B(1/2)__

1/4

1/3

| __B(1/2)__| | | Family -| |

|__G(1/2)__| 

Since each of the remaining three combinations of children is equally likely, the conditional  probability of each combination is 1/3. The conditional probability creates a new revised sample  space, so the probability that a two­child family will include one girl is the sum of the  probabilities of the combinations that include a girl.  Since two of the three members of the sample space include a girl, the probability that the second child is a girl is 1/3 + 1/3 = 2/3. Many people imagine that this probability must be wrong since they think girls should be as likely as boys, but given the information that one child is a boy, a 2/3 probability that

Ask Dr. Math: FAQ 

Boy or Girl?  Note that for purposes of this discussion, we'll assume that having a boy or girl is equally likely, even though statistically that may not be the case for a given population at a given time. In a two-child family, one child is a boy. What is the probability that the other child is a girl? What if the older child is a boy? Does this information change the probability that the second child is a girl? What if we choose the family first? 

When the only information given is that there are two children and one is a boy, here are two ways of  looking at the problem:  Combinations in the Sample Space  In a two­child family, there are four and only four possible combinations of children. We will  label boys B and girls G; in each case the first letter represents the oldest child:  {BB, BG, GB, GG}  When we know that one child is a boy, there cannot be two girls, so the sample space shrinks to:  {BB, BG, GB}  Two of the possibilities in this new sample space include girls:  {BG, GB}  and since there are two combinations out of three that include girls, the probability that the second child is a girl is 2/3.

Probability Tree  We can also visualize this problem using a probability tree.  In the following probability tree, the number next to each letter (B for boy, G for girl) indicates the probability of that event. To calculate the unconditional probability of any one combination of children, we multiply the numbers along that combination's path. Given no information other than that the family has two children, the four combinations of