Probability and Statistics – Mrs. Leahy

Chapter 4: Discrete Probability Distribution ALWAYS KEEP IN MIND: The Probability of an event is ALWAYS between: _____ and _____!!!!

4.1 Probability Distributions Random Variables A random variable is a quantitative (_________________) value associated with an outcome of a probability experiment. Random means x happens by chance. Random variables can be discrete or continuous. In general:

Example 1: Which of the following are continuous and which are discrete. a) Let x = the time it takes a student selected at random to register for the fall term. b) Let x = the number of bad checks drawn on an account on a day selected at random. c) Let x = the amount of gasoline needed to drive your car 200 miles. d) Let x = the number of registered voters, from a sample of 50 registered voters, who voted in the last election

Probability of a Discrete Random Variable A probability distribution shows how “100%” is split among 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?

X 2 3 4 5

P(x) .12 .30 .10 .48

discrete random variables: *the result of a count *a finite or countable number of possibilities *usually an integer

continuous random variables: *the result of a measurement on a continuous scale *an uncountable number of possible outcomes *represented by an interval on a number line

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.

Example 3: Random Experiment: Toss Two Dice Random Variable (X): The sum of the numbers on the dice Probability Distribution (Classical):

Recall that the Law of Large Numbers tells us that if we repeat an experiment enough time that the empirical probability (the actual counts) will approach the classical probability of the event. Probability Distribution (Empirical):

Example 4: Let x = the number of heads when you flip a coin two times. Construct a probability distribution for x. What are the possible outcomes of flipping a coin two times? What are the possible values for x = the number of heads?

2 3 4 5 6 7 8 9 10 11 12

HOW TO CONSTRUCT A PROBABILITY DISTRIBUTION Let x be a discrete random variable with possible outcomes x1, x2, x3, . . . , xn. 1. Make a frequency distribution for the possible outcomes. 2. Find the sum of the frequencies. 3. Find the probability of each outcome by determining its relative frequency (dividing its frequency by the sum of the frequencies). 4. Check that each probability is between 0 and 1 and that the sum of all the probabilities is equal to 1.

Example 5: Use the given frequency distribution to construct a probability distribution and a histogram of the probability distribution. Describe its shape. A company tracks the number of sales per day a new employee makes during a probationary period.

Example 6: Use the given frequency distribution to construct a probability distribution and a histogram of the probability distribution. Describe its shape. A teacher tracks the age of students who email her requesting homework help.

Sales per day, x

Number of days, f

0 16

1 19

2 15

3 21

4 9

5 10

6 9

Age of Student, x

Number of students, f

16 10

17 16

18 20

19 4

Mean and Standard Deviation of a Population Distribution: mean = “expected value of the distribution” standard deviation = “measure of risk”

Let x = the number of correct responses on a four question quiz. Example 7: 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.

Score x

Number of Students

1 3

2 7

3 6

4 4

Formulas:

Expected Value/Mean ( ) ( )E x xP x

Population Prob. Dist.

Standard Deviation 2 2( )x P x

Population Prob. Dist.

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)

Example 8: The table shows the number of times buyers of a product watched a TV infomercial before purchasing the product.

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 9: The table shows how students scored on the last test.

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

Number of times buyers saw infomercial 1 2 3 4 5

Probability 0.27 0.31 0.18 0.09 0.15

Score 55 65 75 85 95

Number of students 2 5 15 10 6

4.2: Binomial Distributions Binomial Experiments If a repeat this experiment several times, I wonder how many times I get this particular outcome? Example 1: Identify n, p, q, and x for each of the binomial experiments below. a) If I roll a die ten times, what is the probability that I get five even numbers? b) If I randomly choose eight students in my SRT, what is the probability that I get six male students in my sample? (Half of the class is male) c) If I spin an eight section spinner five times, what is the probability that I land on “blue” at least twice? (There are 3 blue sections on the spinner) d) If I draw three cards from a standard deck, what is the probability that I get two black cards? e) 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? f) 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? g) 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 no more than four of these people have the blood type B?

Features of a binomial experiment: 1. There are a fixed number of trials/repetitions. Each trial is independent of other trials.

n = the number of trials 2. There are only two outcomes of interest in each trial. Each outcome came be classified as a success (S) or a failure (F). 3. The probability of success is the same in each trial.

p = probability of success = P(S)

q = probability of failure = 1 – p 4. The random variable, x, counts the number of successful trials.

x = the number of successes out

of n trials {0, 1, 2, …, n}

Computing Probabilities for a Binomial Experiment Method 1: Binomial Probability Distribution Formula `Works for ANY binomial probability problem

Example 2: 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 3: Privacy is a concern for many users of the Internet. One survey showed that 70% of Internet users are 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) =

𝑃(𝑥) = 𝐶𝑛,𝑥 ∙ 𝑝𝑥𝑞𝑛−𝑥 =𝑛!

𝑥! (𝑛 − 𝑥)!𝑝𝑥𝑞𝑛−𝑥

n = number of trials

x = number of success out of n trials p = the probability of success

q = 1 – p = the probability of failure

Method 2: Binomial Distribution Table *This table is located in your textbook in Appendix B, Table 2 – Binomial Distribution pg A8, A9, A10* *You will receive a similar table on the test that resembles the handout 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 3 continued) Survey 70% of people are concerned about privacy. You have a sample of 8. What is the probability that 6 will be concerned. c) P(6) = d) P(4 or more) Example 4: 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 5: I roll a die six times. What is the probability that I get a “4” three times? Example 6: You are taking a multiple choice test. There are 4 possibilities per question and there are 12 questions. What is the probability of guessing correctly on: a) 8 questions so you can “pass” the test? b) at least 4 questions? c) at least 1 question? Example 7: (Q#19 pg 211) Fifty-six percent of men do not look forward to going clothes shopping for themselves. You randomly select eight men. Find the probability that the number of men who do not look forward to going clothes shopping is: a) exactly 5 b) more than 6 c) less than 3

Graphing a Binomial Distribution 1. Determine all possible x values for the binomial experiment: x = {0, 1, 2, …, n} 2. Use the Binomial Formula or the Binomial Distribution Table to find P(X) for every value of X. 3. Horizontal Axis - equally spaced bars with “X” at the midpoint. Vertical Axis – Scaled, use P(X)

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 = ______

x P(X)

0

1

2

3

4

5

6

X P(X)

0

1

2

3

4

5

6

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 ). (Example 1 continued) When Jim shoots 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 deviation expected value of x successes out of n trials. of the binomial distribution of the number of Out 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 customers out 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 15 syringes taken from the batch. Suppose the batch contains 5% defective syringes. a) What is the expected number of defective syringes the inspector will find? b) Find σ, the standard deviation. c) What is the probability that the batch will be accepted? Example 4: Sixty-three percent of adults cannot name a Supreme Court Justice. You randomly select five adults and ask them whether they can name a Supreme Court Justice. Let x = the number of adults who cannot name a Supreme Court Justice. a) Find the mean and standard deviation. b) What is the probability that four adults cannot name a Supreme Court Justice.

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4.3: More Discrete Probability Distributions Geometric Distribution I want to repeat an experiment until I have a success. Example: Roll two dice until I get “doubles.” Shoot free-throw shots until I hit one. Keep track of how many green lights I encounter before I finally get stopped at a red one.

Example 1: Basketball player LeBron James makes a free throw shot about 75% of the time. a) Find the probability that the first shot he makes will be on the third attempt. b) Find the probability that he makes his first shot before his fourth attempt. Example 2: Historically, about 60% of the people in your community wear green on St. Patrick’s Day. a) Find the probability that the first person you see wearing green is the fifth person you encounter. b) Find the probability that you see someone wearing green before the third person you encounter.

Features of a geometric experiment: 1. A trial is repeated until a success occurs. Trials are independent of each other. 2. There are only two outcomes of interest in each trial. Each outcome came be classified as a success (S) or a failure (F). 3. The probability of success is the same in each trial.

p = probability of success = P(S)

q = probability of failure = 1 – p 4. The random variable, x, represents the number of the trial in which the first success occurs.

x = the number of the first

successful trial {1, 2, …, n}

The Probability that the first success will occur on trial number x is:

𝑃(𝑥) = 𝑝 ∙ 𝑞𝑥−1

Review – Putting it all together! Probability distribution: Shows all the possible outcome. Each probability is between 0 and 1. The sum of all the probabilities is 1 (100%).

Probability Distribution: Usually starts with a frequency table. Example: I interviewed my some of my students and asked them how many hours of homework they usually have in a night.

You can put probabilities together to answer questions: Q1: What is the probability of at least 3 hours of homework? Q2: What is the probability of at least 1 hour of homework? Q3: What is the probability of 2 or 3 hours of homework?

Expected Value/Mean: Basically a weighted average. Multiply each X value times its probability and then find the sum of all of those. Standard Deviation: Square each x. Multiply by its probability. Find the sum of all those. Subtract the square of your expected value/mean. Square root the answer.

# hours of HW

Frequency f

0 5

1 12

2 14

3 7

4 3

5 2

Expected Value/Mean

( )xP x

Population Prob. Dist. Standard Deviation

2 2( )x P x

Population Prob. Dist.

Binomial Probability distribution: Shows all the possible outcomes of “x successes out of n trials”. Each probability is between 0 and 1. The sum of all the probabilities is 1 (100%).

Binomial Probability: Usually starts with a scenario where there are really only two outcomes: Success and Failure Start by identifying: Number of Trials: n What does “success” mean with respect to the problem?

Probability of success: p What is “failure”? Probability of failure: 1 – p = q Number of “Successes”: x (This will be a number from 0 to n) How to identify the P(x) = (The probability of x successes out of n trials) Method 1: Formula Useful when: only finding the probability of a single “x” value “p” is not a convenient number/not available on the table Method 2: Binomial Probability Distribution Table Useful when: finding several values of P(x), example: “At least 4” , need x = 4, 5, 6, 7 “p” is a convenient number/available in table Example: John has determined that on any given day, he has a probability of wearing the same color shirt as his best friend of 73%. John records the shirt colors of himself and his friend over the next 14 days. What is the probability that he will wear the same color as his friend on 10 of those days? Example: John has determined that on any given day, he has a probability of wearing the same color shirt as his sister of 40%. John records the shirt colors of himself and his sister over the next 8 days. What is the probability that he will wear the same color shirt as his sister on less than 5 of those days?

𝑃(𝑥) = 𝐶𝑛,𝑥 ∙ 𝑝𝑥𝑞𝑛−𝑥

Binomial Probability Distribution: Example: I wrote the names of my students on cards. In my classes 55% of the students are female. I draw 10 names out of a hat. I record the number of names of females.

You can put probabilities together to answer questions: Q1: What is the probability of at least 8 females? Q2: What is the probability of at least 1 female? Q3: What is the probability of 2 or 3 females?

Expected Value/Mean: Multiply the number of trials (n) times the expected success rate (p). Most of the time you will have to round to the nearest whole number for this answer to make sense. Standard Deviation: Multiply the number of trial (n) times the success rate (p) times the failure rate (q). Square root your answer.

Example: Bob’s teacher says that the in the past several years, the probability of passing his final exam is 68%. If there are 36 students enrolled in Bob’s class, what is the expected number of students who will pass the final exam? What is the standard deviation?

# female names X

Probability P(X) You have to calculate this or go get it from the table!

0

1

2

3

4

5

6

7

8

9

10

Expected Value/Mean

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Standard Deviation

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