Problem 39 - Algebra and Trigonometry

Question

Assume equally likely outcomes. Determine the probability of having 1 girl and 3 boys in a 4 -child family.

Step-by-Step Solution

Answer

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

Step 1 - Determine Total Number of Outcomes

Each child can either be a boy (B) or a girl (G). Since there are 4 children, calculate the total number of possible outcomes as follows: \[ 2^4 = 16 \] So, there are 16 possible outcomes.

Step 2 - Identify Desired Outcome

Identify the specific scenario we want: 1 girl (G) and 3 boys (B). This combination can appear in different orders, such as GBBB, BGBB, BBGB, BBGB.

Step 3 - Enumerate Favorable Outcomes

Count the number of ways we can arrange 1 girl and 3 boys. This is a combination problem, which can be calculated using the binomial coefficient: \[ \binom{4}{1} = 4 \] There are 4 ways.

Step 4 - Calculate the Probability

To find the probability, divide the number of favorable outcomes by the total number of outcomes: \[ P(1G \, \text{and} \, 3B) = \frac{4}{16} = \frac{1}{4} \] So, the probability is \( \frac{1}{4} \).

Key Concepts

equally likely outcomes

In probability theory, the idea of 'equally likely outcomes' is fundamental. This means each outcome in a given situation has an equal chance of occurring. For example, when you flip a fair coin, there are two possible outcomes: heads or tails. Since the coin is fair, each outcome is equally likely, giving each a probability of 1/2.
In this exercise, we assume that each child born is equally likely to be a boy or a girl. This assumption is crucial because it allows us to calculate the total number of potential outcomes using simple probability rules.
For 4 children, each of whom could independently be either a boy (B) or a girl (G), the total number of possible combinations is determined by multiplying the number of outcomes for each child. So, with 2 possible outcomes (boy or girl) for each of the 4 children, the total number of outcomes is calculated as follows:

• Each child can be either a boy (B) or a girl (G).
• Total number of outcomes:

\(2^4 = 16\)Thus, there are 16 equally likely outcomes for the family of 4 children.

binomial coefficient

The binomial coefficient, represented as \(\binom{n}{k}\), is a crucial mathematical tool used to find the number of ways to choose \(k\) successes out of \(n\) trials, where the order does not matter. It's a cornerstone in combinatorics and probability theory. For example, in the exercise, we need to determine how many ways we can arrange 1 girl and 3 boys in a family of 4 children.

To calculate \(\binom{4}{1}\), which represents the number of ways to choose 1 girl out of 4 children, we use the following formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Where \(!\) stands for factorial, meaning the product of all positive integers up to that number. So, factorial is defined as:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• \(1! = 1\)
• \((4-1)! = 3! = 3 \times 2 \times 1 = 6\)

Applying the values in the formula, we get:\[\binom{4}{1} = \frac{4!}{1!(4-1)!} = \frac{24}{1 \times 6} = 4\]Therefore, there are 4 ways to arrange 1 girl and 3 boys in a family of 4 children.

combinations in probability

Combinations are essential in probability because they help us count the number of ways events can occur without considering the order. In this case, we use combinations to find the number of ways to have 1 girl and 3 boys in a 4-child family. We already know from the previous section that there are 4 ways to choose 1 girl out of 4 children (\(\binom{4}{1} = 4\)).

Once we know the number of favorable outcomes, we can calculate the probability. To find the probability \(P\) of an event happening, we divide the number of favorable outcomes by the total number of possible outcomes:\[P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}\]In the given problem:

• Number of favorable outcomes (combinations with 1 girl and 3 boys): 4
• Total number of outcomes (possible combinations of 4 children being either boy or girl): 16

Thus, the probability \(P(1G \text{ and } 3B)\) is\[P(1G \text{ and } 3B) = \frac{4}{16} = \frac{1}{4}\]So, the probability of having 1 girl and 3 boys in a 4-child family is \(\frac{1}{4}\).

This approach simplifies complex problems to manageable steps, allowing us to handle real-world scenarios using mathematical precision.