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A tree diagram splits into 2 levels.

Newlyweds Katherine and Matthias plan to have two children. Assuming it is equally likely to be a boy or a girl, use the tree diagram to help you find the probability that their second child is a girl, given that their first child is a boy.

(1 point)
Responses

14
Start Fraction 1 over 4 End Fraction

18
Start Fraction 1 over 8 End Fraction

34
Start Fraction 3 over 4 End Fraction

12
Start Fraction 1 over 2 End Fraction

1 answer

To find the probability that the second child is a girl given that the first child is a boy, we can analyze the possibilities using a tree diagram.

When they have two children, the possible outcomes for the genders of the children are:

  1. Boy, Boy (BB)
  2. Boy, Girl (BG)
  3. Girl, Boy (GB)
  4. Girl, Girl (GG)

Since it's given that the first child is a boy, we only consider the outcomes that start with a boy:

  • Boy, Boy (BB)
  • Boy, Girl (BG)

Now we see what the second child could be in these scenarios:

  1. If the first is a boy, the second could either be a boy (BB) or a girl (BG).

Thus, the outcomes are:

  • BB (first child is a boy, second child is a boy)
  • BG (first child is a boy, second child is a girl)

Now, we want to find the probability that the second child is a girl (BG) given that the first child is a boy.

There are 2 favorable outcomes (1 where the second is a girl) out of 2 possible outcomes (BB and BG):

\[ P(\text{second child is a girl} | \text{first child is a boy}) = \frac{1}{2} \]

So the answer is:

\[ \frac{1}{2} \]

Thus, the correct response is: \( \frac{1}{2} \) (1/2).