Questions: The sample space listing the eight simple events that are possible when a couple has three children is bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg. After identifying the sample space for a couple having four children, find the probability of getting one girl and three boys (in any order). Identify the sample space for a couple having four children. (Use a comma to separate answers as needed.)

The sample space listing the eight simple events that are possible when a couple has three children is bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg. After identifying the sample space for a couple having four children, find the probability of getting one girl and three boys (in any order). Identify the sample space for a couple having four children. (Use a comma to separate answers as needed.)
Transcript text: The sample space listing the eight simple events that are possible when a couple has three children is $\{b b b, b b g, b g b, b g g, g b b, g b g, g g b, g g g\}$. After identifying the sample space for a couple having four children, find the probability of getting one girl and three boys (in any order). Identify the sample space for a couple having four children. $\square$ (Use a comma to separate answers as needed.)
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Solution

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Solution Steps

Step 1: Identify the Sample Space for \(n\) Children

The total number of possible outcomes for \(n=4\) children is \(2^4 = 16\).

Step 2: Calculate the Number of Favorable Outcomes

The number of ways to have \(k=1\) children of a specific gender out of \(n=4\) is given by \(\binom{n}{k} = \frac{n!}{k!(n-k)!} = 4\).

Step 3: Calculate the Probability

The probability of having \(k=1\) children of a specific gender out of \(n=4\) is \(\frac{\binom{n}{k}}{2^n} = 0.25\).

Final Answer:

The probability of having \(k=1\) children of a specific gender when a couple has \(n=4\) children is approximately 0.25.

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