Pre-Calculus Notes — Graphing Rational Functions 2

Statistics Notes — Binomial Probability Experiment

Anytime we make selections from a population without replacement, we do not have independent trials. However, replacement is often not practical. If the number of trials is quite small with respect to the population, we almost have independent trials.
For instance, suppose we select 20 tuition bills at random from a collection of 10,000 bills issued at one college and observe if the bill is in error or not. If 600 of the 10,000 bills are in error, then the probability that the first one selected is in error is 600/10,000, or 0.0600. If the first is in error, then the probability that the second is in error is 599/9999, or 0.0599. Even if the first 19 bills selected are in error, the probability that the 20th is also in error is 581/9981, or 0.0582. All these probabilities round to 0.06, and we can say that the independence condition is approximately satisfied.

Binomial Experiments
There are many probability experiments for which the results of each trial can be reduced to two outcomes: success and failure. For instance, when a basketball player attempts a free throw, he or she either makes the basket or does not.
Probability experiments such as these are called binomial experiments. These experiments are also referred to as Bernoulli experiments, after the Swiss mathematician Jacob Bernoulli.

Features of a binomial experiment
1. There are a fixed number of trials. We denote this number by the letter n.
2. The n trials are independent and repeated under identical conditions.
3. Each trial has only two outcomes: success, denoted by S, and failure, denoted by F.
4. For each individual trial, the probability of success is the same. We denote the probability of success by p and that of failure by q. Since each trial results in either success or failure, p + q = 1 and q = 1 – p.
5. The central problem of a binomial experiment is to find the probability of r successes out of n trials.

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The following notation is helpful, when we talk about binomial probability.

x: The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, …, n
n: The number of trials in the binomial experiment.
p: The probability of success on an individual trial.
q: The probability of failure on an individual trial. (This is equal to 1 – p.)
r: The number of the success of interest
b(r; n, p): Binomial probability – the probability that an n-trial binomial experiment results in exactly r successes, when the probability of success on an individual trial is p.

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Example: From a standard deck of cards, you pick a card, note whether it is a club or not, and replace the card. You repeat the experiment five times.
This is a binomial experiment with success or failure being whether or not a club is selected.
n = 5
p = 13/52 = 1/4
q = 1 – 1/4 = 3/4
x = 0, 1, 2, 3, 4, 5
and, for instance if r = 3, then we would be trying to determine the probability that exactly three of the five cards are clubs.

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Example: Decide whether the experiments are binomial experiments. If they are, specify the values of n, p, q, and x.
1. A certain surgical procedure has an 85% chance of success. A doctor performs the procedure on eight patients.
This is a binomial experiment. There is only success or failure.
n = 8
p = .85
q = .15
x = 0, 1, 2, 3, 4, 5, 6, 7, 8

2. A jar contains five red marbles, nine blue marbles, and six green marbles. You randomly select three marbles from the jar, without replacement.
This is not a binomial experiment. “Without replacement” does not allow each trial to be independent. Also, the sample size is so small so that without replacement, the experiment would not be approximately equal to a binomial experiment.

3. You take a multiple choice quiz that consists of 10 questions. Each question has four possible answers, only one of which is correct. To complete the quiz, you randomly guess the answer to each question. Next time, I suggest you study so you don’t have to randomly guess. Isn’t it a lot more fun when you have a clue about what’s going on. I think so. But anyway.
This is a binomial experiment because you either get each question right or wrong.
n = 10
p = .25
q = .75
x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

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Binomial Probability Formula
In a binomial experiment, the probability of exactly r successes in n trials is P(r) = _nC_r p^r q^n-r

The p^r q^n-r is the probability of getting one outcome with r success and n – r failures. The _nC_r counts the number of outcomes that have r successes and n – r failures.

Example: Privacy is a concern for many users of the internet. One survey showed that 59% of internet users are somewhat concerned about the confidentiality of their email. Based on this information, what is the probability that for a random sample of ten internet users, six are concerned about their privacy?
n = 10
p = .59
q = .41
x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
r = 6

P(6) = _{b(6; 10, .59) = 10}C₆ (.59)⁶ (.41)⁴ = 0.2503034245

In many cases, we will be interested in the probability of a range of successes rather that in the probability of an exact number of successes. For instance, we might wish to compute the probability that at least six of the ten internet users have concerns about their privacy. In such a case, we need to use the addition rule for mutually exclusive events.
P(at least 6 successes) = P(6) + P(7) + P(8) + P(9) + P(10) = 0.6078271782

Example: A biologist is studying a new hybrid tomato. It is known that the seeds of this hybrid tomato have probability 0.70 of germinating. The biologist plants 10 seeds.
What is the probability that exactly eight seeds will germinate?
n = 10
p = .70
q = .30
x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
r = 8

P(8) = ₁₀C₈ (.70)⁸ (.30)² = 0.2334744405

What is the probability that at least eight seeds will germinate?

P(at least 8) = P(8) + P(9) + P(10) = 0.3827827872