Discrete Probability Distributions Chapter(5) Discrete Probability Distributions Note: This PowerPoint is only a summary and your main source should be the book.

Introduction 5-1 Probability Distributions 5-2 Mean , Variance, Standard Deviation ,and Expectation 5-3 The Binomial Distribution Note: This PowerPoint is only a summary and your main source should be the book.

5-1:Probability Distributions A random variable is a variable whose values are determined by chance. Classify variables as discrete or continuous. A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities of the values. Note: This PowerPoint is only a summary and your main source should be the book.

Probability Distribution Table For example 1: S ={T , H} X= number of heads “H” X= 0 , 1 خطوات الحل: ايجاد الحجم الكلي n(S)= (2)^2=4 ايجاد x ايجاد P(X)= عدد ظهور الحدثة \ الحجم الكلي P(x=0) = P(T) = P(x=1) = P(H)= Probability Distribution Table X 1 Total= 1 P(X) Note: This PowerPoint is only a summary and your main source should be the book.

Probability Distribution Table For example 2: S={TT , HT , TH , HH} X= number of heads “H” X= 0 , 1 , 2 = . P(x=0) = P(TT) = P(x=1) = P(HT) + P(TH) = + . = . = . P(x=2) = P(HH)= Probability Distribution Table X 1 2 Total= 1 P(X) Note: This PowerPoint is only a summary and your main source should be the book.

For example 3: TTT TTH THT THH HTT HTH HHT HHH Tossing three coins T H Note: This PowerPoint is only a summary and your main source should be the book.

. . . . S={TTT , TTH , THT , HTT , HHT , HTH , THH , HHH} X= number of heads X= 0 , 1, 2 , 3 . = P(x=0) = P(TTT) = P(x=1) = P(TTH) + P(THT) + P(HTT) = . + P(x=2) = P(HHT) + P(HTH) + P(THH) = . + . = P(x=3) = P(HHH)= Note: This PowerPoint is only a summary and your main source should be the book.

Probability Distribution Table Number of heads (X) 1 2 3 Total= 1 Probability P(X) Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-2: Tossing Coins Represent graphically the probability distribution for the sample space for tossing three coins . X 1 2 3 P(X) Solution : P(X) 2 1 3 X Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-3: The baseball World Series is played by the winner of the National League and the American League. The first team to win four games, wins the world sreies.In other words ,the series will consist of four to seven games, depending on the individual victories. The data shown consist of the number of games played in the world series from 1965 through 2005.The number of games (X) .Find the probability P(X) for each X ,construct a probability distribution, and draw a graph for the data. x Number of games played 4 8 5 7 6 9 16 Note: This PowerPoint is only a summary and your main source should be the book.

Probability Distribution Table P(for 4 games) = = 0.200 P(for 5 games) = = 0.175 P(for 6 games) = = 0.225 P(for 7 games) = = 0.400 Probability Distribution Table X 4 5 6 7 Total= 1 P(X) 0.200 0.175 0.225 0.400 Note: This PowerPoint is only a summary and your main source should be the book.

P(X) 6 4 5 7 0.40 0.30 0.20 0.10 X Note: This PowerPoint is only a summary and your main source should be the book.

The sum of the probabilities of all events in a sample space add up to 1. Each probability is between 0 and 1, inclusively. ∑ p(x) = 1 0 ≤ P(x) ≤ 1 Note: This PowerPoint is only a summary and your main source should be the book.

Determine whether each distribution is a probability distribution. Example 5-4: Determine whether each distribution is a probability distribution. X 5 10 15 20 P(X) √ X 2 4 6 P(X) -1 1.5 0.3 0.2 × X 1 2 3 4 P(X) √ X 2 3 7 P(X) 0.5 0.3 0.4 × Note: This PowerPoint is only a summary and your main source should be the book.

5-2: Mean, Variance, Standard Deviation, and Expectation The mean of a random variable with a discrete probability distribution . Where X1, X2 , X3 ,…, Xn are the outcomes and P(X1), P(X2), P(X3), …, P(Xn) are the corresponding probabilities. µ = X1 . P(X1) + X2 . P(X2) + X3 . P(X3)+ … + Xn . P(Xn) µ = ∑ X . P(X) Note: This PowerPoint is only a summary and your main source should be the book.

Probability Distribution Table Example : Children in Family In a family with two children ,find the mean of the number of children who will be girls. gb g b gg bb bg Solution: Probability Distribution Table X 1 2 Total= 1 P(X) خطوات الحل: ايجاد الحجم الكلي n(S)= (2)^2=4 ايجاد x ايجاد P(X)= عدد ظهور الحدثة \ الحجم الكلي ثم ايجاد الوسط الحسابي mean µ= ∑X . P(X)= 0 . + 1 . + 2 . = 1 Example 5-5, 5-6 : see page 266 Note: This PowerPoint is only a summary and your main source should be the book.

Probability Distribution Table Example 5-7: Tossing Coins If three coins are tossed ,find the mean of the number of heads that occur. Solution: Probability Distribution Table Number of heads (X) 1 2 3 Probability P(X) µ= ∑X . P(X)= 0 . + 1 . + 2 . + 3 . = 1.5 Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-8: No. of Trips of 5 Nights or More The probability distribution shown represents the number of trips of five nights or more that American adults take per year. (That is, 6% do not take any trips lasting five nights or more, 70% take one trip lasting five nights or more per year, etc.) Find the mean. Solution : Note: This PowerPoint is only a summary and your main source should be the book.

Variance and Standard Deviation The formula for the variance of a probability distribution is Variance: Standard Deviation: Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-9: Rolling a Die Compute the variance and standard deviation for the probability distribution in Example 5–5. Solution : Variance standard deviation Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-10: Selecting Numbered Balls A box contains 5 balls .Two are numbered 3 , one is numbered 4 ,and two are numbered 5. The balls are mixed and one is selected at random . After a ball is selected , its number is recorded . Then it is replaced . If the experiment is repeated many times , find the variance and standard deviation of the numbers on the balls. Solution : Number on each ball (X) 3 4 5 Probability P(X) Note: This PowerPoint is only a summary and your main source should be the book.

Step 1 : Step 2 : µ= ∑X . P(X)= 3 . + 4 . + 5 . = 4 Variance 3 4 5 Number on each ball (X) 3 4 5 Probability P(X) X2 32=9 42=16 52=25 Step 1 : µ= ∑X . P(X)= 3 . + 4 . + 5 . = 4 Variance Step 2 : standard deviation Note: This PowerPoint is only a summary and your main source should be the book.

What the value K would be needed to complete the probability Days 0 1 2 3 4 probability 0.06 0.7 0.2 0.03 0.01 A) Mean = 1.23, variance= 0.4171 B) Mean = 0.645, variance = 1.23 C) Mean =1.23, variance= 1.93 D) Mean =1.93, variance = 1.23 6 4 2 1 X 0.2 0.3 K 0.1 P(x) What the value K would be needed to complete the probability distribution? A) 0.15 B) 0.2 C) -0.25 D) -0.2 What is the probability value of x less than or equal 2

1- X is a discrete random variable 1- X is a discrete random variable. The mean of its probability distribution is 3, and the standard deviation is 4. Find ∑X2 P(X). 2- A box contains 6 balls. One is numbered 2, three are numbered 3 and two are numbered 4. Construct a probability distribution for the numbers on the balls.

Expectation The expected value, or expectation, of a discrete random variable of a probability distribution is the theoretical average of the variable. The expected value is, by definition, the mean of the probability distribution. Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-12: Winning Tickets One thousand tickets are sold at $1 each for a color television valued at $350. What is the expected value of the gain if you purchase one ticket? Solution : Win Lose Gain(X) $349 -$1 Probability P(X) An alternate solution : Note: This PowerPoint is only a summary and your main source should be the book.

Example : Winning Tickets One thousand tickets are sold at $1 each for four prizes of $100, $50, $25, and $10. After each prize drawing, the winning ticket is then returned to the pool of tickets. What is the expected value if you purchase two tickets? Gain X Probability P(X) $98 $48 $23 $8 -$2 Solution : Note: This PowerPoint is only a summary and your main source should be the book.

An alternate solution : Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-13: Six balls numbered 1,2,3,5,8 and 13 are placed in a box Example 5-13: Six balls numbered 1,2,3,5,8 and 13 are placed in a box. A ball is selected at random, and its number is recorded and then it is replaced. Find the expected value of the numbers that will occur.

1-If a player rolls one die and when gets a number greater than 4, he wins 12$, the cost to play the game is 5$. What is the expectation of the gain? A) 2$ B) -1$ C) -2$ D) 1$ 2- One thousand tickets are sold at $3 each for a PC value at $1600. What is the expected value of the gain if a person purchases two ticket?

EX (page:273)

Exercises 5-1 page 263-264 7-8-9-10-11-12-13-14-15-16-17-18 Exercises 5-2 page 272-273 1-2-10-15-18

Mean, Variance and Standard deviation for The Binomial Distribution Many types of probability problems have only two possible outcomes or they can be reduced to two outcomes. Examples include: when a coin is tossed it can land on heads or tails, when a baby is born it is either a boy or girl, etc. Note: This PowerPoint is only a summary and your main source should be the book.

Each trial can have only two possible outcomes—success or failure. The binomial experiment is a probability experiment that satisfies these requirements: Each trial can have only two possible outcomes—success or failure. There must be a fixed number of trials. The outcomes of each trial must be independent of each other. The probability of success must remain the same for each trial. Example 5-15: see page 276-277 Note: This PowerPoint is only a summary and your main source should be the book.

Notation for the Binomial Distribution P(S) P(F) p q P(S) = p n X Note that X = 0, 1, 2, 3,...,n :The symbol for the probability of success :The symbol for the probability of failure :The numerical probability of success :The numerical probability of failure and P(F) = 1 – p = q :The number of trials :The number of successes Note: This PowerPoint is only a summary and your main source should be the book.

In a binomial experiment, the probability of exactly X successes in n trials is Note: This PowerPoint is only a summary and your main source should be the book.

The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution

Example 5-16: Tossing Coins A coin is tossed 3 times. Find the probability of getting exactly two heads. n = 3 x= 2 p= Solution : Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-17: Survey on Doctor Visits A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. n = 10 x= 3 p= Solution : Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-18: Survey on Employment A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part- time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs. n = 5 x= 3,4,5 p=0.30 q=1-0.30 =0.70 Solution : Q: find the probability that at most 3 of them will have part-time jobs.

Mean, Variance and Standard deviation for the binomial The mean , variance and SD of a variable that the binomial distribution can be found by using the following formulas: Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-22: Tossing A Coin A coin is tossed 4 times. Find the mean, variance and standard deviation of number of heads that will be obtained. Solution : n = 4 p= Note: This PowerPoint is only a summary and your main source should be the book.

Example 5-23: Rolling a die A die is rolled 360 times , find the mean , variance and slandered deviation of the number of 4s that will be rolled . n = 360 p= Solution : Note: This PowerPoint is only a summary and your main source should be the book.

A coin is tossed 72 times. The standard deviation for the number of heads that will be tossed is A student takes a 6 question multiple choice quiz with 4 choices for each question. If the student guesses at random on each question, what is the probability that the student gets exactly 3 questions correct? A) 0.088 B) 0.0512 C) 0.132 D) 0.022

A student takes a 6 question multiple choice quiz with 4 choices for each question. If the student guesses at random on each question, what is the probability that the student gets exactly 3 questions wrong? How many times a die is rolled when the mean of the numbers greater than 4 that will be rolled = 20? If 7% of calculators are defective, find the mean of the number of defective calculators for a lot of 4200 calculators?

Which of the following is a binomial experiment? Asking 100 people if they swim. B) Testing five different brands of aspirin to see which brands are effective. C) Asking 60 people which brand of soap they buy. D) Drawing four balls without replacement from a box contains 5 white balls, 7 blue balls and one green ball.

Exercises 5-3 page 283-284-285 1-5-6-11-13-17-20-26 Review Exercises page 304-305 1-2-3-8-16-17 Chapter Quiz page 306-307 All except 19-20 and 25 to 33