Section 2.4 Bernoulli Trials

A bernoulli trial is a repeated experiment with the following properties:

1. There are two outcomes of each trial: “success” and “failure.”

2. The probability of success in each trial is the same.

3. The trials are independent of each other.

1. Which of the following experiments are bernoulli trials?

(a) Rolling a fair die 20 times and, each time, observing if a 2 appears.

(b) Selecting 4 cards from a standard deck of 52 cards without replacement and, each time, observing if an ace is drawn.

(c) Assuming that there are 5 yellow and 7 white marbles in a jar. Choosing three marbles from the jar with replacement (put the marble back in the jar) and, each time, observing if a yellow marble is choosen.

Random Variable A random variable is a rule that assigns a number to each outcome of a chance experiment.

The random variable X in this section is associated with the outcomes of the bernoulli experiments. Computing Probabilities in a Bernoulli Experiment: Exactly x successes In a bernoulli experiment in which the probability of a success in any trial is p, the probability of exactly x successes in n independent trials is given by

P (X = x) = C(n, x)px(1 − p)n−x Computing Probabilities in a Bernoulli Experiment: At Most x successes In a bernoulli experiment in which the probability of a success in any trial is p, the probability of at most x successes in n independent trials is given by

P (X ≤ x) = P (X = 0) + P (X = 1) + ...P (X = x)

Bernoulli experiments are the same as Binomial experiments, and so we will use the Calcula- tor function for the binomial distribution to compute the probabilities associated with bernoulli experiments.

Calculator Steps:

2ND , VARS , scroll down to A or click ALPHA , MATH . Your screen should show binompdf( , this is for exactly x successes. For at most x successes, scoll down to B or click ALPHA , APPS . This time your screen should show binomcdf( .

Here’s the format for both: binompdf(n, p, x) or binomcdf(n, p, x) .

2. It is estimated that one third of the general population has blood type A+. A sample of six people is selected at random. (Round answers to four decimal places.)

(a) What is the probability that exactly two of them have blood type A+?

(b) What is the probability that at most two of them have blood type A+?

2 Spring 2018, © Maya Johnson 3. The probability that a DVD player produced by VCA Television is defective is estimated to be 0.06. A sample of ten players is selected at random. (Round answers to four decimal places.)

(a) What is the probability that the sample contains no defective units?

(b) What is the probability that the sample contains at most two defective units?

A Few Diﬀerent Ways to use binomcdf

1. P (X ≥ x) = 1− binomcdf(n, p, x − 1) “at least x successes.”

2. P (x1 ≤ X ≤ x2) = binomcdf(n, p, x2)− binomcdf(n, p, x1 − 1) “at least x1 but at most x2 successes.”

3. P (x1 < X ≤ x2) = binomcdf(n, p, x2)− binomcdf(n, p, x1) “more than x1 but at most x2 successes.”

4. P (x1 ≤ X < x2) = binomcdf(n, p, x2 −1)− binomcdf(n, p, x1 −1) “at least x1 but fewer than

x2 successes.”

5. P (x1 < X < x2) = binomcdf(n, p, x2 − 1)− binomcdf(n, p, x1) “more than x1 but fewer than

x2 successes.”

4. From experience, the manager of Kramer’s Book Mart knows that 50% of the people who are browsing in the store will make a purchase. What is the probability that among ten people who are browsing in the store, at least four will make a purchase? (Round answer to four decimal places.)

3 Spring 2018, © Maya Johnson 5. Suppose 30% of the restaurants in a certain part of a town are in violation of the health code. A health inspector randomly selects ﬁve of the restaurants for inspection. (Round answers to four decimal places.)

(a) What is the probability that none of the restaurants are in violation of the health code?

(b) What is the probability that one of the restaurants is in violation of the health code?

6. The manager of Toy World knows that the probability an electronic game will be returned to the store is 0.22. If 54 games are sold in a given week, determine the probabilities of the following events. (Round answers to four decimal places.)

(a) No more than 12 games will be returned.

(b) At least 8 games will be returned.

4 Spring 2018, © Maya Johnson 7. A coin is biased so that the probability of tossing a head is 0.46. If this coin is tossed 54 times, determine the probabilities of the following events. (Round answers to four decimal places.)

(a) The coin lands heads more than 21 times.

(b) The coin lands heads fewer than 28 times.

8. A company ﬁnds that one out of four employees will be late to work on a given day. If this company has 40 employees, ﬁnd the probabilities that the following number of people will get to work on time. (Round answers to four decimal places.)

(a) Exactly 31 workers or exactly 35 workers.

(b) At least 27 workers but fewer than 35 workers.