Problem 37 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The probability is \( \frac{1}{8} \).

Step 1: Determine Total Possible Outcomes

First, calculate the total number of possible outcomes for a 3-child family. Each child can be either a boy (B) or a girl (G). Therefore, the total number of possible outcomes is given by the combinations of B and G over 3 children. This results in: \[ 2^3 = 8 \] The possible outcomes are: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG.

Step 2: Determine Favorable Outcomes

Next, identify the outcomes where there are exactly 3 boys. The only combination that meets this criteria is BBB. Hence, there is 1 favorable outcome.

Step 3: Calculate Probability

Finally, calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes: \[ P(\text{3 boys}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{8} \]

Key Concepts

Equally Likely Outcomes

In probability, equally likely outcomes mean that each possible result of an event has the same chance of occurring. Imagine flipping a fair coin: it has two equally likely outcomes, heads or tails.
When we talk about equally likely outcomes for a 3-child family, each child has an equal chance of being a boy (B) or a girl (G).
This assumption of equally likely outcomes simplifies our calculations, as every scenario (like BBB or GGG) has the same chance of happening.

Combinations

Combinations help us understand how different groups can form. For example, in a family with 3 children, each can either be a boy (B) or a girl (G).
This gives us several possible combinations of genders.
We calculate it as \(2^3 = 8\), showing all possible groupings (like BBB, BBG, etc.).
The idea of combinations is crucial in figuring out total possible outcomes and finding specific groups among those combinations.

Favorable Outcomes

Favorable outcomes refer to the specific results that meet our criteria for an event. For instance, if we want exactly 3 boys in a 3-child family, BBB is our favorable outcome.
Out of all the possible combinations (BBB, BBG, etc.), we only consider outcomes that fit what we're looking for.
In this case, we count only 1 favorable outcome (BBB).

Total Possible Outcomes

Total possible outcomes count all the different results that could occur from an event. For a 3-child family, each child can be a boy or girl.
Hence, with two choices per child over 3 children, we calculate \[2^3 = 8\ \] different outcomes, like BBB, BGB, etc.
Listing all these outcomes helps us understand the full range of possibilities, making it easier to see the chance of a specific favorable outcome occurring.