Problem 21 - Algebra for College Students

Question

A couple plans to have 3 children. Assuming males and females are equally likely, what is the probability that they have either 3 boys or 3 girls?

Step-by-Step Solution

Answer

The probability is \( \frac{1}{4} \).

Step 1: Determine Total Possible Outcomes

Each child can either be a boy (B) or a girl (G). With three children, there are a total of 2^3 = 8 possible outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG.

Step 2: Identify Favorable Outcomes

The favorable outcomes for having either 3 boys or 3 girls are: BBB and GGG.

Step 3: Count the Favorable Outcomes

There are 2 favorable outcomes (BBB and GGG).

Step 4: Calculate the Probability

The probability is the number of favorable outcomes divided by the total possible outcomes. Therefore, P(3 boys or 3 girls) = \( \frac{2}{8} \) = \( \frac{1}{4} \).

Key Concepts

favorable outcomes

In probability, 'favorable outcomes' refer to the specific outcomes of an event that match the condition or conditions we are interested in. For example, in this exercise, we are interested in the scenario where a couple has either 3 boys or 3 girls. The favorable outcomes are the sequences of births that meet these criteria. So, the favorable outcomes are 'BBB' for 3 boys, and 'GGG' for 3 girls.
Make sure to understand that favorable does not necessarily mean 'good,' but rather the outcomes that match the event you’re calculating the probability for.

total possible outcomes

The total possible outcomes are all the different ways an event can occur. For this problem, each child can either be a boy or a girl, giving us two options per child. Since the couple has 3 children, the number of ways to arrange 3 children with each being either a boy or a girl is given by the formula for combinations, which is 2 raised to the power of the number of children: \[ 2^3 = 8 \].
The 8 possible combinations (total outcomes) are: BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG.

probability calculation

Probability measures the likelihood of a specific event to occur out of all possible events. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. To find the probability that a couple has either 3 boys or 3 girls:\[ P = \frac{number \text{ of favorable outcomes}}{total \text{ possible outcomes}} \].
We counted 2 favorable outcomes (BBB and GGG) and 8 total possible outcomes. So we calculate:
\[ P(3 \text{ boys or 3 girls}) = \frac{2}{8} = \frac{1}{4} \].
Therefore, the probability is \( \frac{1}{4} \), or 25%.

binomial outcomes

Binomial outcomes are situations where there are two possible outcomes for each trial. In this exercise, each child can be either a boy or a girl. With 3 children, we have multiple binomial outcomes. The binomial distribution helps us to calculate probabilities in such scenarios by considering the number of trials, the number of successes, and the probability of success for each trial.
In standard binomial problems (like this one), the probability of each child being a boy (success) or a girl (failure) is 0.5. The formula to calculate any specific binomial probability is given by:\[ P(X=k) = {n \text{ choose } k} \times p^k \times (1-p)^{n-k} \],
where \(n\) is the total number of trials, \(k\) is the number of successes, and \(p\) is the probability of success on each trial. However, for simplicity, in this exercise, we're directly counting the outcomes.