One family has 4 children. The probability of having at least 2 boys is 12/16, or 3/4. Group of answer choices True False
The correct answer and explanation is:
Answer: True
To determine if the probability of having at least 2 boys in a family with 4 children is 1216\frac{12}{16} or 34\frac{3}{4}, let’s first consider all possible gender combinations for the four children. Each child has two possible outcomes: boy (B) or girl (G). The total number of possible outcomes for 4 children is 24=162^4 = 16.
Next, we calculate the number of outcomes where there are at least 2 boys. This means we want to find all outcomes where there are 2, 3, or 4 boys.
- 2 boys: The number of ways to have 2 boys and 2 girls is (42)=6\binom{4}{2} = 6. This can be written as choosing 2 spots for boys out of 4.
- 3 boys: The number of ways to have 3 boys and 1 girl is (43)=4\binom{4}{3} = 4. This is choosing 3 spots for boys out of 4.
- 4 boys: The number of ways to have 4 boys is (44)=1\binom{4}{4} = 1. There is only one way this can occur.
So, the total number of outcomes where there are at least 2 boys is:
6+4+1=116 + 4 + 1 = 11
Therefore, the probability of having at least 2 boys is the ratio of favorable outcomes (11) to total possible outcomes (16), which is 1116\frac{11}{16}, not 1216\frac{12}{16}.
The statement is False. The correct probability is 1116\frac{11}{16}.