15C - Sampling with replacement

Bernoulli trials and sequences

When objects are replaced after their initial selection in a sampling experiment the probability of each event remains constant. When this is the case we can analyse the sampling as a Bernoulli sequence (named after Daniel Bernoulli).

A Bernoulli sequence has the following properties:

1. The experiment consists of n independent trials (often called a Bernoulli trial).
2. The result of each trial has only two outcomes: "success" and "failure".
3. As the trials are independent the probability of success, p, remains constant. The probability of failure, 1 - p, also remains constant.

The binomial distribution

The binomial distribution is the probability distribution relating to Bernoulli sequences. In general, the probability of achieving x successful events in n independent trials of a binomial experiment is (x = 0, 1, 2, ..., n):

15C - VIDEO EXAMPLE 1:

Sarah is a keen cyclist. The probability that she rides her bike on any given day is 0.6. What is the probability that Sarah rides her bike exactly 4 times in 1 week? State your answer correct to 2 decimal places.

Cumulative probability

To determine the probability that X take a value in the interval a and b (including the values of a and b), the values of p(x) from x = a to x = b are added together.

Binomial distributions using CAS

CAS calculators are extremely efficient at calculating probabilities associated with Bernoulli trials. The commands are located under Interactive → Distributions/Inv. Dist → Discrete.

BinomialPDf

The command binomialPDf is used to determine the probability of an exact number of successes, x, in n trials.

15C - VIDEO EXAMPLE 2:

Jenny has a scoring percentage of 75% in netball. What is the probability that she scores exactly 5 goals if she has 12 scoring opportunities? State your answer correct to 2 decimal places.

BinomialCDf

The command binomialCDf is used to determine the probability when the number of successes, x, lies within an interval [a, b] inclusive.

15C - VIDEO EXAMPLE 3:

Jesse is a basketball player with a free throw shooting percentage of 82%. If he has 10 free throw shooting opportunities in a match, calculate the probability that he scores 7 or more goals. State your answer correct to 2 decimal places.

Finding sample sizes (n) using CAS

In some cases you may be given the probability of success and asked to find the sample size (n) that results in a certain probability. Such questions may be phrased as "for what value of n does the probability become …”. This is a more difficult process and you need different techniques for solving, such as:

• Solving numerically.
• Solving graphically.
• Defining a function and solving by trail and error.

These techniques will be demonstrated in the examples below.

15C - VIDEO EXAMPLE 4:

The probability of winning a game is 0.45. What is the least number of games that must be played to ensure that the probability of winning at least twice is more than 0.95?

15C - VIDEO EXAMPLE 5:

Anne is playing an easy course of Mini Golf. Her probability of getting a hole in one is 0.6. What is the minimum number of shots needed for the probability of Anne getting a hole in one exactly five times to be more than 25%?