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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 34 Start Fraction 3 over 4 End Fraction 18 Start Fraction 1 over 8 End Fraction 14 Start Fraction 1 over 4 End Fraction 12 Start Fraction 1 over 2 End Fraction

1 answer

To find the probability that Katherine and Matthias's second child is a girl, given that their first child is a boy, we can represent all possible outcomes using a tree diagram.

Let's denote:

  • "B" for Boy
  • "G" for Girl

The possible outcomes for two children are:

  1. First child is a Boy, second child is a Boy (BB)
  2. First child is a Boy, second child is a Girl (BG)
  3. First child is a Girl, second child is a Boy (GB)
  4. First child is a Girl, second child is a Girl (GG)

The outcome combinations based on their first child being a boy will be:

  • If the first child is B, the outcomes are:
    1. First is B, Second is B (BB)
    2. First is B, Second is G (BG)

Thus, the two relevant cases are:

  • BB (First is a Boy, Second is also a Boy)
  • BG (First is a Boy, Second is a Girl)

Now we need to find the probability that the second child is a Girl (G) given that the first child is a Boy (B).

From our outcomes, we have:

  • 1 favorable outcome (BG)
  • 2 total possible outcomes (BB and BG)

Therefore, the probability that their second child is a girl given that their first child is a boy is:

\[ P(\text{second child is G} | \text{first child is B}) = \frac{\text{Number of favorable outcomes (BG)}}{\text{Total outcomes given first child is B (BB, BG)}} = \frac{1}{2} \]

Thus, the answer is:

\(\frac{1}{2}\)