Lesson Objectives At the end of the lesson, students can: Recognize and describe the 4 attributes of a binomial distribution. Use binompdf and binomcdf commands Determine a binomial coefficient Determine the mean and standard deviation of a binomial distribution

Binomial Distributions The Binomial Setting – a situation where the following 4 conditions are satisfied … (remember BINS)   1. 2. 3. 4. Binary? The possible outcomes of each trial can be classified as “success” or “failure”. Independent? Does knowing the result of one trial must not have any effect on the result of any other trial. Number? The number of trials n of the chance process must be FIXED in advance. Success? On each trial, the probability p of success must be the same.

Binomial Distributions * If data are produced in a binomial setting, then the random variable X = number of successes is called a binomial random variable.   * The distribution of X in the binomial setting is the binomial distribution with parameters n and p. NOTATION: X is _______________ n = ________________________ p = ________________________ possible values of X are the whole #s 0 to n. # of successes # of trials in the chance process probability of success on any one trial

Binomial Distributions One of the lowest scoring AP Free Response Test questions are on binomial distribution because often students do not recognize that using the binomial distribution is appropriate. So let’s practice identifying binomial distributions. Remember, when you are having trouble answering a probability question, check to see if it is a binomial setting! (BINS)

Binomial Distributions Examples: Is it reasonable to use a binomial distribution as a model for the following situations?   A basketball player makes 68% of his free throws.. During a particular game, he shoots 10 free throws. X = # of shots made. Binary? Yes success = make a shot Independent? Yes, it is reasonable to assume that making one shot does not change the probability of making another. Number? Yes, there are 10 free throws Success? Yes, he has a 68% chance each time. This is a binomial setting. X is a binomial random variable with parameters n = 10 and p = 0.68

Binomial Distributions Examples: Is it reasonable to use a binomial distribution as a model for the following situations?   The probability of having the blood type O from a particular set of parents is 0.25. The couple has 5 children. X = number of children with blood type O. Binary? Yes success = blood type O Independent? Yes, children inherit genes determining blood type independently from their parents. Number? Yes, there are 5 children Success? Yes, the probability of a “success” is 0.25 . This is a binomial setting. X is a binomial random variable with parameters n = 5 and p = 0.25

Binomial Distributions Examples: Is it reasonable to use a binomial distribution as a model for the following situations?   Deal 10 cards from a shuffled deck of 52 cards. X = # of red cards. Binary? Yes success = # of red cards Independent? No, since you are not replacing the cards, each card’s probability is affected by the card dealt before it. This is a NOT binomial setting.

MORE PRACTICE Examples: Is it reasonable to use a binomial distribution as a model for the following situations?   Observe the next 100 cars that go by and let C = color. Binary? No, there are more than two possible colors. Also, C is not even a random variable since the outcomes aren’t numerical. This is not a binomial setting.

MORE PRACTICE Binary? Yes success = the number of sixes Examples: Is it reasonable to use a binomial distribution as a model for the following situations?   Roll a fair die 10 times and let X = the number of sixes. Binary? Yes success = the number of sixes Independent? Yes, the die is fair and one roll does not affect the probability of the next roll. Number? Yes, rolling the die 10 times Success? Yes, the probability of a “success” is 1/6 or 0.167. This is a binomial setting. X is a binomial random variable with parameters n = 10 and p = 0.167.

MORE PRACTICE More practice examples on p. 384 in book and Check Your Understanding (answers in back of book on p. 385)

Binomial Formulas 5•4•3•2•1 = 120 1 Binomial Formulas – these should be used in place of your calculator when the problem specifies!   *Binomial coefficient – the number of ways of arranging k successes among n observations is:  NOTE: 𝑛 𝑘 is read “binomial coefficient n choose k”, (which is the formula for a combination--aka nCr in your calculator!) ! is factorial notation; 8! is read “8 factorial” 5! = ________________ on your calc MATH  PRB  4: ! 0! = _______ 5•4•3•2•1 = 120 1

Binomial Formulas Flip a coin 5 times. Consider getting a “heads” a success. Determine the number of different arrangements of 3 successes (3 heads) among 5 observations (5 coin tosses). Or you can put in your calculator Under the Math menu  PRB  option 3: nCr 5 nCr 3 = 10

Binomial Formulas *Binomial probability – If X has the binomial distribution with n observations and probability p of success on each observation, the possible values of X are 0, 1, 2, …, n. If k is any one of these values, then: This is the binompdf function in the calc – this formula is on the formula sheet, so you just need to know when and how to use it!

Binomial Formulas Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that exactly 2 of them have type O blood? There is a 26.37% chance that exactly 2 children have O type blood.

Binomial Formulas Suppose you purchase a bundle of 10 bare-root broccoli plants. The sales clerk tells you that on average you can expect 5% of the plants to die before producing any broccoli. Assume that the bundle is a random sample of plants. Use the binomial formula to find the probability that you will lose at most one of the broccoli plants. Note that you can check your answer in your calculator using the binomcdf(10,.05, 1)

Binomial Formulas The probability of having the blood type O from a particular set of parents is 0.25. The couple has 5 children. X = number of children with blood type O. Use the binomial probability formula to find the probability that at least one of the children in this example has blood type O. B(5, 0.25) There is a 76.27% that at least one of the children will have blood type O.

Binomial Probabilities Finding Binomial Probabilities – you’ve calculated these probabilities by hand, and now, we’ll use our calculators!   TI-84: binompdf (n, p, X)  found under 2nd DISTR / 0:binompdf    “pdf” stands for probability distribution function. If X is a discrete random variable, the pdf assigns a probability to each value of X. Please note that on the AP Free Response Exam, you will not receive much credit for just showing the calculator technique. At the very least, you must indicate what each of those calculator inputs represent. How to show complete work will be in the future slides.

Binomial Probabilities Corinne is a basketball player who makes 75% of her free throws over the course of a season. In a particular game, Corinne shoots 12 free throws. What is the probability that she makes exactly 8 of the 12 shots? (Check BINS first!) P(X = 8) = Binompdf(12,0.75,8) = 0.1936 Corinne has a 19.36% probability of making 8 of the 12 free throws.

Binomial Probabilities A quality engineer selects an SRS of 10 switches from a large shipment for detailed inspection. Unknown to the engineer, 10% of the switches in the shipment fail to meet the specifications. What is the probability that no more than 1 of the 10 switches in the sample fail inspection? P(X ≤ 1) = P(X = 0) + P(X = 1) = Binompdf(10,0.10,0) + Binompdf(10,0.10,1) = 0.7361 There is a 73.61% chance of no more than 1 of the 10 switches in the sample to fail inspection.

Binomial Probabilities Corinne is a basketball player who makes 75% of her free throws over the course of a season. In a particular game, Corinne shoots 12 free throws. What is the probability that she makes at most 8 of the 12 shots? P(X ≤ 8) = P(X=0)+ P(X=1) + . . . +P(X=8) = Binompdf(12,0.75,0) + Binompdf(12,0.75,1) + Binompdf(12,0.75,2) + Binompdf(12,0.75,3) + Binompdf(12,0.75,4) + Binompdf(12,0.75,5) + Binompdf(12,0.75,6) + Binompdf(12,0.75,7) + Binompdf(12,0.75,8) = 0.3512

Binomial Probabilities *Oftentimes, we want to find the probability that a random variable takes a range of values (problem #2 and 3) as opposed to a specific value (#1). The cumulative binomial probability is useful in these cases.   Cumulative distribution function (cdf) of random variable X calculates the sum of the probabilities for 0, 1, 2, …, up to the value X. In other words, it calculates the probability of obtaining at most X success in n trials. TI-84: binomcdf (n, p, X)  found under 2nd DISTR / A:binomcdf

Binomial Probabilities Corinne is a basketball player who makes 75% of her free throws over the course of a season. In a particular game, Corinne shoots 12 free throws. What is the probability that she makes at most 8 of the 12 shots? Use the cumulative distribution function. P(X≤8) = Binomcdf(12,0.75,8) = 0.35122 There is a 35.12% probability that Corinne will make at most 8 of the 12 shots.

Binomial Probabilities A quality engineer selects an SRS of 10 switches from a large shipment for detailed inspection. Unknown to the engineer, 10% of the switches in the shipment fail to meet the specifications. X = # of defective switches, B(10, 0.1)   probability that there is at most 3 defective switches probability that there are more than 3 defective switches    probability that there are more than 4 defective switches P(X≤3)=Binomcdf(10, 0.1, 3) = 0.9872 P(X>3)= 1 - Binomcdf(10, 0.1, 3) = 0.0128 P(X>4)= 1 - Binomcdf(10, 0.1, 4) = .0016

Binomial Formulas μx = np σx = 𝑛𝑝(1−𝑝) Binomial Mean and Standard Deviation – these formulas only work for binomial distributions!! Check BINS!   If a count X has the binomial distribution with number of observations n and probability of success p, then the mean and standard deviation of X are: Mean: Standard Deviation: These are also in your formula packet!!! μx = np σx = 𝑛𝑝(1−𝑝)

Binomial Formulas .30 15 µ = 15(.30) = 4.5 A factory employs several thousand workers, of whom 30% are women. If the 15 members of the union executive committee were chosen from the workers at random, the number of women on the committee would have the binomial distribution with n = _ _______ and p = ________.   Find the mean number of women on a randomly chosen committee of 15 workers. What is the standard deviation of the count X of women members on the committee? .30 15 µ = 15(.30) = 4.5 This means of a randomly selected committee of 15 members, we would expect there to be between 4 and 5 women members σ = 𝟏𝟓(.𝟑𝟎)(𝟏−𝟎.𝟑𝟎) =1.77 This means of the randomly selected 15 members, the number of women members would differ from 4.5 by an average of 1.77.

Check answers to odd problems Homework Read Textbook pages p. 382 – 393 Do exercises p. 403 – 404 #71 – 73, 75 – 78, 80, 82, 84 Check answers to odd problems

Lesson Objectives At the end of the lesson, students can: Recognize and describe the 4 attributes of a binomial distribution. Use binompdf and binomcdf commands Determine a binomial coefficient Determine the mean and standard deviation of a binomial distribution