Problem 43 - Algebra 2

Question

Genetics A family has five children. Assume that the probability of having a boy is \(0.5 .\) Write the term in the expansion of \((b+g)^{5}\) for each outcome described. Then evaluate each probability. \(\begin{array}{llll}{\text { a. exactly } 3 \text { boys }} & {\text { b. exactly } 4 \text { boys }} & {\text { c. exactly } 4 \text { girls }}\end{array}\)

Step-by-Step Solution

Answer

The probability of having exactly 3 boys is 0.3125; for having exactly 4 boys it is 0.15625; and for having exactly 4 girls it is also 0.15625.

Step 1: Apply the binomial theorem for exactly 3 boys

We need to find the coefficient of \(b^3g^2\) in the expansion of \((b+g)^5\) for exactly 3 boys & 2 girls. According to the binomial theorem, this is given by \({5 \choose 3} b^{3} g^{2} = 10 * b^{3} * g^{2}\). The probability of this outcome is \(10 * (0.5)^{3} * (0.5)^{2}=0.3125\).

Step 2: Apply the binomial theorem for exactly 4 boys

We need to find the coefficient of \(b^4g^1\) in the expansion of \((b+g)^5\) for exactly 4 boys & 1 girl. According to the binomial theorem, this is given by \({5 \choose 4} b^{4} g^{1} = 5 * b^{4} * g^{1}\). The probability of this outcome is \(5 * (0.5)^{4} * (0.5)^{1}=0.15625\).

Step 3: Apply the binomial theorem for exactly 4 girls

We need to find the coefficient of \(b^1g^4\) in the expansion of \((b+g)^5\) for exactly 1 boy & 4 girls. According to the binomial theorem, this is given by \({5 \choose 1} b^{1} g^{4} = 5 * b^{1} * g^{4}\). The probability of this outcome is \(5 * (0.5)^{1} * (0.5)^{4}=0.15625\).

Key Concepts

Probability

Probability is the measure of how likely an event is to occur. It is a fundamental concept in statistics and mathematics.
For this exercise, we are looking at the probability of having a certain number of boys among five children. Probability values range between 0 and 1, where 0 means the event will not happen and 1 means it will certainly happen.

The probability for a single event, like having a boy, can be understood simply as:

• If the chance of having a boy child is 0.5, the probability of having a girl is also 0.5. This is because these two outcomes (boy or girl) are equally likely.
• To find the probability of multiple independent events happening, like having 3 boys and 2 girls in five births (independent events), we multiply the probabilities for each event.

Probability is crucial because it allows us to determine the likelihood of complex outcomes by constructing and evaluating these probability models.

Probability Distribution

Probability distribution shows all the possible outcomes of a random experiment and assigns a probability to each outcome. It provides a complete description of the randomness of an event.
For example, when dealing with genetic outcomes like boys and girls, probability distributions tell us all the different ways these children could be boys or girls, and how likely each scenario is.

Consider this:

• When we say there is a distribution for having boys in a family of five, each possible number of boys (from 0 to 5) has a specific probability attached to it.
• The most common statistical distributions include normal distribution, binomial distribution, and uniform distribution. Here, we use a binomial distribution.

A binomial probability distribution is used when we have a fixed number of independent trials, each with the same probability of success (e.g., having a boy). Understanding probability distributions helps us model real-world behaviors and uncertainties.

Binomial Expansion

Binomial expansion is a way to expand expressions that are raised to a power, such as \( (b + g)^5 \). It is instrumental in solving probability problems involving combinations of two possibilities, like flipping a coin or using scenarios in genetics.

The binomial expansion formula is given by: \[ (a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{(n-k)} b^{k} \] Here, \( {n \choose k} \) is a binomial coefficient that tells you how many ways you can choose \( k \) successes from \( n \) trials, and \( a^{(n-k)} b^{k} \) are the terms of the expansion.

In this context:

• For 3 boys and 2 girls, we look at the term \( b^3g^2 \), and the binomial coefficient \( {5 \choose 3} \) yields 10 corresponding scenarios.
• Each term's coefficient multiplies with the respective probabilities \( (0.5)^3 \, (0.5)^2 \).

Using binomial expansion simplifies calculations in probability, especially in predicting outcomes with multiple possible results.