Probability and Statistics – Mrs. Leahy

Unit 5: The Binomial Probability Distribution and Related Topics

Section 1: Introduction to Random Variables and Probability Distributions(Day 1)Random Variables

A random variable is a quantitative (_________________)

variable “x” if the value of “x” in a given experiment or

observation is a _______________ or _________________

outcome. Random variables can be discrete or continuous.

In general:

Example 1:Which of the following are continuous and which are discrete.

a) the time it takes a student selected at random to register for the fall term.

b) the number of bad checks drawn on an account on a day selected at random.

c) the amount of gasoline needed to drive your car 200 miles.

d) the number of registered voters, from a sample of 50 registered voters, who voted in the last election

e) the number of pencils left on the table

f) the time it takes for a boy to run 100 meters

g) the number of history majors in a random sample of 100 college students

h) the weight of a firefighter, which must be between 150 – 200 lbs.Probability of a Discrete Random Variable

discrete random variables: *the result of a count*can take on only a finite number of values or a countable number of values*usually a whole number

continuous random variables:*the result of a measurement on a continuous scale*can take on any of countless variables in a range*usually a whole number or a decimal

A probability distribution is an assignment of probabilities to each distinct value of a discrete random variable or to each interval of values of a continuous random variable.

Example 2:

Is this a valid probability distribution?

Example: Random Experiment: Toss Two CoinsRandom Variable (X): Count the number of Heads

Example:Random Experiment: Toss Two DiceRandom Variable (Y): The sum of the numbers on the dice

Probability Distribution:

Recall that the Law of Large Numbers tells us that if we repeat an experiment enough time that the relative frequency (the actual counts) will eventually equal the theoretical probability of the event. So if we have a VERY large same of numbers the histogram that represents the data should look very much like the probability histogram of the data!

Mean and Standard Deviation of a Population Distribution:

mean = “expected value of the distribution”

standard deviation = “measure of risk”

Features of the probability distribution of a discrete random variable

1. EACH distinct value is assigned a probability (recall: 0 ≤ P(x) ≤ 1)2. The sum of all the assigned probabilities must be 1.

Formulas:

Expected Value/Mean Population Prob. Dist.

Standard Deviation Population Prob. Dist.

X 2 3 4 5P(x) .1

2.30 .1

0.48

Probability Distribution

X 0 1 2P(x)

.25 .50 .25

Example 3: A quiz with possible score 1, 2, 3, and 4 was given to a class of 20 students. The score distribution is listed in the table to the left.

a) Make a probability distribution b) Find the mean, μ, and standard table for this distribution deviation, σ, for the distribution

c) Make a graph for this probability distribution.

Example 4: The table shows the number of times buyers of a product watched a TV infomercial before purchasing the product.

Can we use this as a valid probability distribution?

If you randomly select a student from this class,

d) P(2)

e) Find P(at most 2)

f) Find P(at least 2)

g) P(3 or 4)

Scorex

Number ofStudents

1 32 73 64 4

Number of times buyers saw infomercial 1 2 3 4 5Percentage of buyers 27% 31% 18% 9% 15%

a) Graph the probability distribution. b) Compute the expected value μ and the standard deviation σ

c) If a buyer is selected at random, find the following:

P(saw infomercial 3 or more times) P(saw infomercial 1 or 2 times)

Example 5: At a carnival, you pay $1.00 to play a coin-flipping game with two fair coins. If you get Heads on both coins then you win $3.00.

a) What is the variable of interest? x =

b) What is the sample space for the random variable?

c) What are the four outcomes of throwing two coins?

d) What are your expected earnings? (μ) e) Compare the cost to play vs. the expected earnings

Simulation

We can use a random number table to simulate the outcomes from a given discrete probability distribution.

earning

frequency P(x) x P(∙x)

03

THERE IS A RANDOM NUMBER TABLE IN YOUR TEXTBOOKAppendix G, pg A-101 , about 3 pages from the back of the book

Example 6: John hits at home run at 20% of the time at bat.Let x be the random variable that counts the number of home runs out of 10 attempts at bat. Consider the numbers 0 to 9 of the random number table.

Let 0 an 1 represent “home runs”Let 2 through 9 represent “not home run”

a) Start at line 5, column 3 of the random number table. Determine the results of 10 trials. How many “home runs” occurred in this simulation?

b) What if 0 through 7 represent “not home run” and 8 & 9 represent “home run” Using the same results as (a), how many “home runs” occurred in this simulation.

Example 7: Use the random number table. Start at line 7, column 5.

Let an even number = heads, odd number = tails

What is the probability of getting heads in twenty trials?

Section 2: Binomial Probabilities(Day 2)

Binomial Experiments

As the number of trials increase, the Probability from the simulation will approach the true probability 0.2

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 the same circumstances.

3. Each trial has only two-outcomes: success (S) and failure (F).

4. For each individual trial, the probability of success is the same. P(success) = p P(failure) = qp + q = 1 and 1 – p = q

5. The central problem in a binomial experiment is to find the probability of r successes out of n trials.

Flip a coin 2 times and count the number of heads.

Fixed number of trials:Two outcomes: Success = p(success) =

Failure = p(failure) =

Find the Probability of getting 2 heads if you flip a coin 2 times: Example: On a game show, a contest gets a chance to try his hand at spinning a roulette wheel called the “wheel of fortune.” The wheel is divided into 36 slots, only one of which is painted gold. If the ball lands in the gold slot, the contestant wins $50,000. No other place on the wheel pays.

What is the probability that the quiz show will have to pay the fortune to 3 contestants out of 100?

Is this a binomial experiment?

a. Is there a fixed number of trials? If so, what is n?

b. Are the trials independent?

c. How many outcomes are in the trial? What are they?

d. In each trial, what is the probability for S or F?

e. Find the probability of ____ successes out of ____ trials. What are n, r, p, and q?

Example 1: Consider this problem: throw a fair die 10 times and find the probability that the number 3 appears twice. Does this fit the definition of a binomial experiment?

a) Is there a fixed number of trials? If so, what is n?

b) Are the trials independent?

c) How many outcomes are in the trial? What are they?

d) In each trial, what is the probability for S or F?

e) Find the probability of _____ successes out of _____ trials. What are n, r, p, and q?

Example 2: In a certain college, 60% of students live on campus. Suppose we randomly choose 20 students from this college as ask whether each of them live on campus. What is the probability that 5 live on campus?

a) Let’s say “live on campus” is a success. What is the P(Success) ?

b) What is the definition for a “failure”? What is the P(Failure) ?

c) In this experiment, what is the number of trials? n = ______

d) We are looking for the probability that 5 of the 20 students live on campus. In this case r = _____

Example 3: According to the Textbook of Medical Physiology, 9% of the population has blood type B. Suppose we choose 18 people at random from the population and test the blood type of each. What is the probability that four of these people have the blood type B?

a) What makes up a trial? What is a success? What is a failure?

b) What are the values of n, p, q, and r?

Computing Probabilities for a Binomial Experiment

Method 1: Binomial Probability Distribution FormulaWorks for ANY binomial probability problem

P (r )= n!r ! (n−r )!

prqn−r=Cn ,r ∙ prqn−r

n = number of trialsr = number of success out of n trialsp = the probability of successq = 1 – p = the probability of failure

Example 4: You throw a fair die 10 times. Find the probability that the number 3 appears two times.

Find the probability that the number 3 appears at most two times.

Example 5: Privacy is a concern for many users of the Internet. One survey showed that 70% of Internet users are somewhat concerned about the confidentiality of their e-mail. Based on this information, what is the probability that for a random sample of 8 Internet users, 6 are concerned about the privacy of their e-mail.

a) What is n, p, q, and r ? b) P(6) =

Method 2: Binomial Distribution Table

Works only on binomial probability problems where p = nice friendly values listed in the table….If it’s NOT in the table, you HAVE to use method 1 on the previous notes page.

Here is a sample (small piece) of binomial distribution table:

(Example 5 continued)

b) P(6) = c) P(4 or more)

Example 6: A college conducts a survey and finds that 45% of students call home every weekend. Ten random students are selected.

a) What is the probability that exactly 5 students will say they call home every weekend.

b) What is the probability that all the students will say they call home every weekend?

c) What is the probability that fewer than four students will say they call home every weekend?

Example 7: I roll a die six times. What is the probability that I get a “4” three times?

Section 3: Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

Example 1: Jim enjoys playing basketball. He figures that he makes about 50% of the field goals he attempts during a game. Make a histogram showing the probability that Jim will make 0,1,2,3,4,5 or 6 shots out of six attempted field goals.

n = ______

success =

p = ______

**USE TABLE OR FORMULA TO FILL THIS IN!!!***

Example 2: A waiter at the Green Spot Restaurant has learned from long experience that the probability that a lone diner will leave a tip is only 0.7. During one lunch hour, the waiter serves 6 people who are dining by themselves. Make a graph of the binomial probability distribution that show the probabilities that 0,1,2,3,4,5, or 6 lone diners leave tips.

n = ______

success =

p = ______

Mean and Standard Deviation of a Binomial Distribution

In the case of a binomial distribution, there is a special way to calculate the balance point (the mean μ) and the measure of spread (the standard deviation σ ).

r P(r)0123456

r P(r)0123456

μ=npσ=√npq

(Example 1 continued) When Jim shots field goals in basketball games, the probability that he makes a shot is only 0.5.

The mean of the binomial distribution is the For six trials, what is the standard deviationexpected value of r successes out of n trials. of the binomial distribution of the number ofOut of 6 throws, what is the expected number successful field goals Jim makes?of goals that Jim will make?

(Example 2 continued) The waiter at the Green Spot Restaurant expects that the probability of a lone diner leaving a tip is only 0.7.

What is the mean of this binomial distribution? What is the standard deviation?The mean is the represents the number of customersout of 6 that he expects to leave a tip.

Example 3: The quality control inspector of a production plant will reject a batch of syringes if two or more defective are found in a random sample of eight syringes taken from the batch. Suppose the batch contains 1% defective syringes.

a) Make a histogram showing the probabilities of r = 0, 1, 2, 3, 4, 5, 6, 7, and 8 defective syringes in a random sample of eight syringes.b) Find μ. What is the expected number of defective syringes the inspector will find?c) What is the probability that the batch will be accepted?d) Find σ.