Constructing and Interpreting a Probability Distribution for Discrete Random Variables

Q6. The random variable x represents the number of boys in a family of three children. Assuming that boys and girls are equally likely, (a) construct a probability distribution, and (b) graph the probability histogram.

Background

Topic: Discrete Probability Distributions & Binomial Distribution

This question tests your understanding of how to construct a probability distribution for a discrete random variable, specifically using the binomial distribution. It also asks you to represent the distribution visually as a histogram.

Key Terms and Formulas

• Discrete Random Variable: A variable that can take on a countable number of distinct values.

• Binomial Distribution: Used when there are a fixed number of independent trials, each with two possible outcomes (success/failure).

• Probability Mass Function for Binomial:

$P(x) = \frac{n!}{x!(n-x)!} p^x (1-p)^{n-x}$

• $n$ = number of trials (children = 3)

• $x$ = number of successes (number of boys)

• $p$ = probability of success (boy = 0.5)

Step-by-Step Guidance

1. Identify the possible values for $x$: Since there are three children, $x$ can be 0, 1, 2, or 3 (number of boys).

2. Use the binomial formula to calculate $P(x)$ for each value of $x$:

   • For $x = 0$: $P(0) = \frac{3!}{0!3!} (0.5)^0 (0.5)^3$

   • For $x = 1$: $P(1) = \frac{3!}{1!2!} (0.5)^1 (0.5)^2$

   • For $x = 2$: $P(2) = \frac{3!}{2!1!} (0.5)^2 (0.5)^1$

   • For $x = 3$: $P(3) = \frac{3!}{3!0!} (0.5)^3 (0.5)^0$

3. Calculate each probability value using the formula above, but do not compute the final numeric values yet.

4. Once you have the probabilities, organize them into a table with $x$ and $P(x)$.

5. Draw a probability histogram using the values from your table. The histogram should have $x$ (number of boys) on the horizontal axis and $P(x)$ (probability) on the vertical axis.

Try solving on your own before revealing the answer!