# DETERMINE whether the conditions for a binomial setting are met. COMPUTE and INTERPRET probabilities involving binomial random variables

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1 1 Section 7.B Learning Objectives After this section, you should be able to DETERMINE whether the conditions for a binomial setting are met COMPUTE and INTERPRET probabilities involving binomial random variables CALCULATE the mean and standard deviation of a binomial random variable and INTERPRET these values in context Binomial Settings 2 When the same chance process is repeated several times, we are often interested in whether a particular outcome does or doesn t happen on each repetition. In some cases, the number of repeated trials is fixed in advance and we are interested in the number of times a particular event (called a success ) occurs. If the trials in these cases are independent and each success has an equal chance of occurring, we have a binomial setting. Definition: A binomial setting arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome occurs. The four conditions for a binomial setting are B I N S Binary? The possible outcomes of each trial can be classified as success or failure. Independent? Trials must be independent; that is, 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. 1

2 3 4 2

3 From Blood Type to Aces (continued) 6 From Blood Type to Aces (continued) 3

4 Binomial Random Variable 7 Consider tossing a coin n times. Each toss gives either heads or tails. Knowing the outcome of one toss does not change the probability of an outcome on any other toss. If we define heads as a success, then p is the probability of a head and is 0. on any toss. The number of heads in n tosses is a binomial random variable X. The probability distribution of X is called a binomial distribution. Definition: The count X of successes in a binomial setting is a binomial random variable. The probability distribution of X is a binomial distribution with parameters n and p, where n is the number of trials of the chance process and p is the probability of a success on any one trial. The possible values of X are the whole numbers from 0 to n. Note: When checking the Binomial condition, be sure to check the BINS and make sure you re being asked to count the number of successes in a certain number of trials! Binomial Probabilities Example 1 8 In a binomial setting, we can define a random variable (say, X) as the number of successes in n independent trials. We are interested in finding the probability distribution of X. Type O Example Each child of a particular pair of parents has probability 0.2 of having type O blood. Genetics says that children receive genes from each of their parents independently. These parents have children. What is the probability that none of the children has type O blood? Binomial Example The count X of children with type O blood is a binomial random variable with n = trials and probability p = 0.2 of a success on each trial. It is reasonable to assume that each child s blood type is independent of each other. In this setting, a child with type O blood is a success (S) and a child with another blood type is a failure (F). What s P(X = 0)? 4

5 Example 1 1) What is the probability that none of the children has type O blood? P(Success) = (0.2) P(Failure) = (0.7) P(X=0) = P(FFFFF) = (0.7)(0.7)(0.7)(0.7)(0.7) = (0.7) =.2373 There is about a 24% chance that none of the children have type O blood. 9 Binomial Example Note: There is only one possible outcome in this scenario to have none of the children have type O blood 2) What is the probability that ONE of the children has type O blood? 10

6 11 2) What is the probability that EXACTLY ONE of the children has type O blood (continued)? 3) What is the probability that EXACTLY TWO of the children have type O blood? 12 Probability of 2 with Type O blood P(SSFFF) = (0.2)(0.2)(0.7)(0.7)(0.7) = (0.2) 2 (0.7) 3 = However, there are a number of different arrangements in which 2 out of the children have type O blood: SSFFF SFSFF SFFSF SFFFS FSSFF FSFSF FSFFS FFSSF FFSFS FFFSS Verify that in each arrangement, P(X = 2) = (0.2) 2 (0.7) 3 = Therefore, P(X = 2) = 10(0.2) 2 (0.7) 3 = There is about a 26% chance that 2 of the children have type O blood. 6

7 Binomial Coefficient Note, in the previous example, any one arrangement of 2 S s and 3 F s had the same probability. This is true because no matter what arrangement, we d multiply together 0.2 twice and 0.7 three times. We can generalize this for any setting in which we are interested in k successes in n trials. That is, Definition: P(X k) P(exactly k successes in n trials) = number of arrangements p k (1 p) nk The number of ways of arranging k successes among n observations is given by the binomial coefficient n n! k k!(n k)! for k = 0, 1, 2,, n where n! = n(n 1)(n 2) (3)(2)(1) and 0! = Binomial Probability The binomial coefficient counts the number of different ways in which k successes can be arranged among n trials. The binomial probability P(X = k) is this count multiplied by the probability of any one specific arrangement of the k successes. Binomial Probability B(n,p) If X has the binomial distribution with n trials and probability p of success on each trial, the possible values of X are 0, 1, 2,, n. If k is any one of these values, P(X k) n p k (1 p) nk k 14 Number of arrangements of k successes Probability of k successes Probability of n-k failures 7

8 8 1 Binomial Probabilities Example 3 (an easier way to find the number of possible outcomes) 16 Now, Find These Binomial Coefficients

9 Example 4 - Using the Binomial Probability Formula 17 Type O Blood Example B(,.2) Let X = the number of children with type O blood. (a) Find the probability that exactly 3 of the children have type O blood. P(X 3) (0.2) 3 (0.7) 2 10(0.2) 3 (0.7) (b) Should the parents be surprised if more than 3 of their children have type O blood? To answer this, we need to find P(X > 3). P(X 3) P(X 4) P(X ) 4 (0.2) 4 (0.7) 1 (0.2) (0.7) 0 (0.2) 4 (0.7) 1 1(0.2) (0.7) Since there is only a 1.% chance that more than 3 children out of would have Type O blood, the parents should be surprised! Example 4 Using the Calculator to find Binomial Probabilities Answer Example 4 using the TI84 Always State B(,.2) 18 9

10 EXAMPLE : Expected Value and Expected Variance of a Binomial Distribution We describe the probability distribution of a binomial random variable just like any other distribution by looking at the shape, center, and spread. Describe our probability distribution for: X = number of children with type O blood in a family with children. x i p i Shape: The probability distribution of X is skewed to the right. It is more likely to have 0, 1, or 2 children with type O blood than a larger value. 19 Center: The median number of children with type O blood is 1. Based on our formula for the mean (find the expected value): X x i p i (0)(0.2373) 1(0.391)... ()( ) 1.2 Spread: The variance of X is 2 X (x i X ) 2 p i (0 1.2) 2 (0.2373) (11.2) 2 (0.39)... ( 1.2) 2 ( ) The standard deviation of X is X Mean and Standard Deviation of a Binomial Distribution Notice, the mean µ X = 1.2 can be found another way. We can use the parameters n and p; and the method below: Mean and Standard Deviation of a Binomial Random Variable If a count X has the binomial distribution with number of trials n and probability of success p, the mean and standard deviation of X are X np X np(1 p) 20 Note: These formulas work ONLY for binomial distributions. They can t be used for other distributions! 10

11 EXAMPLE 6: Mean and Standard Deviation of a Binomial Distribution Type O Example: X = number of children with type O blood in a family with children. B(,.2) 21 Find the mean and standard deviation of X. Since X is a binomial random variable with parameters n = and p =.2, we can use the formulas for the mean and standard deviation of a binomial random variable. X np (.2) 1.2 X np(1 p) 4(.2)(.7).968 We d expect at least 1 of the children to have Type O blood, on average. If this was repeated many times with groups of children, the number with Type O blood would differ from 1.2 children by an average of 1 child. 22 Summary In this section, we learned that A binomial setting consists of n independent trials of the same chance process, each resulting in a success or a failure, with probability of success p on each trial. The count X of successes is a binomial random variable. Its probability distribution is a binomial distribution. The binomial coefficient counts the number of ways k successes can be arranged among n trials. If X has the binomial distribution with parameters n and p, the possible values of X are the whole numbers 0, 1, 2,..., n. The binomial probability of observing k successes in n trials is P(X k) n p k (1 p) nk k 11

12 23 Summary In this section, we learned that The mean and standard deviation of a binomial random variable X are X np X np(1 p) 12

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