Question

Assume equally likely outcomes. Determine the probability of having 3 girls in a 3 -child family.

Short answer

\( P(\text{3 girls}) = \frac{1}{8} \)

Step-by-step solution

Understand the Problem

Identify that there are 3 children, and we are seeking the probability that all 3 are girls. Each child can either be a girl or a boy.

Determine Possible Outcomes

List all the possible outcomes of having 3 children. The outcomes are: GGG, GGB, GBG, GBB, BGG, BGB, BBG, BBB, where 'G' stands for girl and 'B' stands for boy.

Count Favorable Outcomes

Identify the favorable outcome for having 3 girls, which is only 'GGG'.

Calculate Total Outcomes

Count the total number of outcomes. There are 8 outcomes in total.

Find the Probability

Calculate the probability of the favorable outcome. Probability is the number of favorable outcomes divided by the total number of outcomes: \[ P(\text{3 girls}) = \frac{1}{8} \]

Key concepts

Sample Space

Understanding the concept of Sample Space is crucial when tackling probability problems. The sample space consists of all possible outcomes in a given scenario. In our example, with three children, the outcomes include every possible combination of boys (B) and girls (G).

To create the sample space, we list all combinations of having three children:

GGG (all girls), GGB, GBG, GBB, BGG, BGB, BBG, and BBB (all boys). This gives us a total of 8 different outcomes. Knowing the sample space helps us understand both the number and nature of the possible events.

Favorable Outcomes

Next, we need to identify which outcomes are Favorable Outcomes for the event we're interested in. In this case, the event is having exactly three girls. This means we are looking for the outcome GGG within our sample space.

So, among the 8 possible outcomes, only one is favorable: GGG. Understanding which outcomes are favorable is key to calculating probabilities, as it directly affects the probability fraction.

Total Outcomes

When dealing with probability, knowing the Total Outcomes is equally important. This refers to the number of all potential results in the sample space. In our example, since each child can be either a boy (B) or a girl (G), we have a total of 2 choices per child.

Therefore, the total number of outcomes for 3 children is calculated as follows: \[2 \times 2 \times 2 = 2^3 = 8\]
We count 8 total outcomes when listing all the possible combinations of boys and girls.

Probability Calculation

Finally, let’s look at the actual Probability Calculation. Probability is defined as the ratio of the number of favorable outcomes to the number of total outcomes. Using our earlier figures, we know:

• Number of Favorable Outcomes (only GGG): 1
• Total Number of Outcomes: 8

So, the probability \[P(\text{3 girls}) = \frac{1}{8}\]
This means there is a 1 in 8 chance, or a probability of 12.5%, that a family with 3 children will have all girls.