Questions: Consider a family with 4 children. Assume the probability that one child is a boy is 0.5 and the probability that one child is a girl is also 0.5, and that the events "boy" and "girl" are independent. (a) List the equally likely events for the gender of the 4 children, from oldest to youngest. (Let M represent a boy (male) and F represent a girl (female), Select all that apply?) FMMF MFMM FFMF MFFF FMFM FMMM MMMF FFFF FFMM MFFM FMFF MMMM MMFF MFMF MMFM FFFM (b) What is the probability that all 4 children are male? (Enter your answer as a fraction.) Notice that the complement of the event "all four children are male" is "at least one of the children is female," Use this information to compute the probability that at least one child is female: (Enter your answer as a fraction.)

Solution Steps

To solve this problem, we need to consider the possible gender combinations for a family with 4 children, where each child has an independent probability of 0.5 of being male (M) or female (F).

(a) We list all possible combinations of M and F for 4 children. Since each child can be either M or F, there are \(2^4 = 16\) possible combinations.

(b) To find the probability that all 4 children are male, we calculate the probability of the event "MMMM". Since each child being male is independent and has a probability of 0.5, the probability of all 4 being male is \(0.5^4\).

(c) The probability that at least one child is female is the complement of the probability that all children are male. We subtract the probability of all being male from 1.

To determine the possible gender combinations for a family with 4 children, where each child can be either male (M) or female (F), we calculate all permutations of M and F for 4 positions. Since each child has 2 possible outcomes, there are \(2^4 = 16\) combinations. These combinations are:

\[ \begin{align_} &\text{MMMM}, \text{MMMF}, \text{MMFM}, \text{MMFF}, \text{MFMM}, \text{MFMF}, \text{MFFM}, \text{MFFF}, \\ &\text{FMMM}, \text{FMMF}, \text{FMFM}, \text{FMFF}, \text{FFMM}, \text{FFMF}, \text{FFFM}, \text{FFFF} \end{align_} \]

The probability that each child is male is \(0.5\). Since the events are independent, the probability that all 4 children are male (MMMM) is:

\[ P(\text{MMMM}) = 0.5^4 = \frac{1}{16} \]

The probability that at least one child is female is the complement of the probability that all children are male. Therefore, we subtract the probability of all being male from 1:

\[ P(\text{at least one F}) = 1 - P(\text{MMMM}) = 1 - \frac{1}{16} = \frac{15}{16} \]

Final Answer