Questions: You select a family with four children. If M represents a male child and F a female child, the set of equally likely outcomes for the children's genders is shown below. Find the probability of selecting a family with at least one female child. FFFF, FFFM, FFMF, FMFF, MFFF, MFFM, MFMF, MMFF, FFMM, FMFM, FMMF, FMMM, MFMM, MMFM, MMMF, MMMM The probability of having at least one female child is . (Type an integer or a simplified fraction.)

$You select a family with four children. If M represents a male child and F a female child, the set of equally likely outcomes for the children's genders is shown below. Find the probability of selecting a family with at least one female child. FFFF, FFFM, FFMF, FMFF, MFFF, MFFM, MFMF, MMFF, FFMM, FMFM, FMMF, FMMM, MFMM, MMFM, MMMF, MMMM The probability of having at least one female child is . (Type an integer or a simplified fraction.)$

Transcript text: You select a family with four children. If $M$ represents a male child and $F$ a female child, the set of equally likely outcomes for the children's genders is shown below. Find the probability of selecting a family with at least one female child. \{FFFF, FFFM, FFMF, FMFF, MFFF, MFFM, MFMF, MMFF, FFMM, FMFM, FMMF, FMMM, MFMM, MMFM, MMMF, MMMM\} The probability of having at least one female child is . $\square$ (Type an integer or a simplified fraction.)

Solution

Solution Steps

Step 1: Total Outcomes

The total number of possible outcomes for the genders of four children is given by $ 2^4 = 16 $. This accounts for each child being either male ($ M $) or female ($ F $).

Step 2: All Male Outcome

There is only one specific outcome where all children are male, which is represented as $ MMMM $. Thus, the number of outcomes with all male children is $ 1 $.

Step 3: Favorable Outcomes

To find the number of favorable outcomes (families with at least one female child), we subtract the all-male outcome from the total outcomes: \[ \text{Favorable Outcomes} = \text{Total Outcomes} - \text{All Male Outcome} = 16 - 1 = 15 \]

Step 4: Probability Calculation

The probability of selecting a family with at least one female child is calculated by dividing the number of favorable outcomes by the total number of outcomes: \[ P(\text{at least one female}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{15}{16} = 0.9375 \]