Binomial Distribution

Concepts

• Bernoulli trials: A Bernoulli trial is an experiment with only two possible outcomes: success (usually denoted as 1) or failure (usually denoted as 0).

• Probability of success: The probability of success in a single Bernoulli trial is denoted as p.

• Number of trials: The binomial distribution considers a fixed number of independent Bernoulli trials, denoted as n.

• Probability mass function: The probability mass function (PMF) of the binomial distribution calculates the probability of obtaining a specific number of successes in n trials.

AP Style Example Questions

1. What is the probability of getting exactly 3 successes in 5 Bernoulli trials with a success probability of 0.6?

2. If the probability of success is 0.2 and there are 8 trials, what is the probability of getting at least 2 successes?

Answers to the Example Questions

1. To calculate the probability of getting exactly 3 successes in 5 trials, we can use the binomial PMF formula. The probability is given by:
Substituting the values, we get:
Therefore, the probability is approximately 0.2592.

2. To calculate the probability of getting at least 2 successes in 8 trials, we need to calculate the probabilities of getting 2, 3, 4, ..., 8 successes and sum them up. Using the binomial PMF formula, we can calculate each probability and add them together.
After performing the calculations, we find that the probability is approximately 0.9898.

AP Style Exercises

1. A fair coin is tossed 10 times. What is the probability of getting exactly 7 heads?

2. In a basketball game, a player has a free throw success rate of 0.75. If the player takes 5 free throws, what is the probability of making at least 3 of them?

Answers to the Exercises

1. Using the binomial PMF formula, the probability of getting exactly 7 heads in 10 coin tosses is approximately 0.1172.

2. To calculate the probability of making at least 3 free throws out of 5 with a success rate of 0.75, we need to calculate the probabilities of making 3, 4, and 5 free throws and sum them up. After performing the calculations, we find that the probability is approximately 0.9819.