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Part 1
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 three

girls

and one

boy left parenthesis in any order right parenthesis
.
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Part 1
Identify the sample space for a couple having four children.
​{enter your response here
​}
​(Use a comma to separate answers as​ needed.)

### Part 1: Sample Space for a Couple Having Four Children

When a couple has four children, each child can either be a boy (B) or a girl (G). The sample space will list all the possible combinations of four children. Since each child has two possible outcomes, the total number of combinations is \( 2^4 = 16 \).

Here is the sample space for four children:

**Sample Space**:  
\[
\{ bbbb, bbbg, bbgb, bbgg, bgbb, bgbg, bggb, bggg, gbbb, gbbg, gbgb, gbgg, ggbb, ggbg, gggb, gggg \}
\]

### Part 2: Probability of Getting Three Girls and One Boy (in any order)

Now, we want to calculate the probability of having three girls and one boy, in any order. 

First, we identify the favorable outcomes from the sample space where there are exactly three girls and one boy. These are:

**Favorable Outcomes**:  
\[
\{ bggg, gbgg, ggbg, gggb \}
\]

There are 4 favorable outcomes. Since the total number of possible outcomes is 16, the probability \( P \) is calculated as:

\[
P(\text{three girls and one boy}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{4}{16} = \frac{1}{4}
\]

### Final Answer:
The probability of having three girls and one boy (in any order) is \( \frac{1}{4} \).

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Would you like more details or have any questions? Here are 5 related questions:

1. How would the sample space change if the couple had five children?
2. What is the probability of getting two boys and two girls?
3. How does the probability change if we assume boys and girls are not equally likely?
4. What is the probability of getting all boys?
5. How many different outcomes are possible for three children?

**Tip**: When finding probabilities, always check how many total outcomes are possible in the sample space first.

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 three girls and one boy (in any order).

Learn how to identify the sample space for a couple having four children and calculate the probability of getting three girls and one boy in any order. This problem explores key probability concepts, combinatorics, and sample space. Suitable for students in Grades 6-8.

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