Problem 21 - Algebra and Trigonometry

Question

Genders of Children The ratio of male to female births is in fact not exactly one-to-one. The probability that a newborn turns out to be a male is about \(0.52 .\) A family has 10 children. (a) What is the probability that all 10 children are boys? (b) What is the probability all are girls? (c) What is the probability that 5 are girls and 5 are boys?

Step-by-Step Solution

Answer

(a) Probability for all boys: \(0.52^{10}\). (b) Probability for all girls: \(0.48^{10}\). (c) Probability for 5 boys and 5 girls: \(\binom{10}{5} \times (0.52)^5 \times (0.48)^5\).

Step 1: Identify Probability Values

We are given that the probability of a newborn being a male is \( p = 0.52 \). Consequently, the probability of a newborn being female is \( 1 - p = 0.48 \).

Step 2: Calculate Probability for All Boys

To find the probability that all 10 children are boys, we use the formula for probability of independent events: \[P(\text{all boys}) = 0.52^{10}. \]Calculate this to find the probability of all being boys.

Step 3: Calculate Probability for All Girls

Similarly, to calculate the probability that all children are girls, we use:\[P(\text{all girls}) = 0.48^{10}.\]Calculate this to find the probability of all being girls.

Step 4: Calculate Probability for 5 Boys and 5 Girls

For a mixture of 5 boys and 5 girls, use the binomial probability formula:\[P(5 \text{ boys and } 5 \text{ girls}) = \binom{10}{5} \times (0.52)^5 \times (0.48)^5,\]where \(\binom{10}{5}\) is the binomial coefficient, or "10 choose 5." Calculate each component to find this probability.

Key Concepts

Binomial Probability

Binomial probability deals with scenarios where there are only two possible outcomes for each trial. In the exercise, the outcomes are boy and girl. The situation involves a fixed number of trials — here, the family has ten children. Binomial probability tells us the likelihood of achieving a specific number of successes (or a specific outcome) out of these trials.

When applying binomial probability, we use the formula: \[P(k) = \binom{n}{k} \, p^k \, (1-p)^{n-k},\]where:

• \(n\) is the total number of trials (or children, which is 10 in this case).
• \(k\) is the number of successes for which we're calculating the probability. Here this could be the number of boys or girls we want (e.g., 5 boys out of 10 children).
• \(p\) is the probability of success (boy is 0.52, and girl is 0.48).

Understanding the binomial probability concept is crucial for calculating the chances of having exactly 5 boys and 5 girls, or all boys or all girls.

Independent Events

In probability, independent events are those whose outcomes do not affect each other. For example, whether the first child is a boy or a girl doesn't change the probability of gender for any following children. Each child's gender is an independent event.

This concept was used in the exercise to calculate the probability of all 10 children being boys or girls. We know that the probability of a child being a boy is 0.52. Thus, for each independent birth, this probability remains the same. So the probability that all 10 children are boys becomes \(0.52^{10}\), and for all girls, it's \(0.48^{10}\). Each calculation is simply multiplying the probability of one child by itself for each of the independent events with no deviation.

Probability Distribution

Probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. In the exercise, we could create a probability distribution for the number of boys in 10 births. This distribution would include all possible outcomes from 0 to 10 boys.

For example:

• 0 boys (and thus 10 girls)
• 1 boy and 9 girls
• 5 boys and 5 girls
• Up to 10 boys

Each outcome comes with its own probability, calculated using the methods outlined in the binomial probability section. Creating a probability distribution allows one to see the most and least likely outcomes quickly. It's a comprehensive visualization of the potential results and their probabilities.