6.3: The Binomial Model

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1 6.3: The Binomial Model The Normal distribution is a good model for many situations involving a continuous random variable. For experiments involving a discrete random variable, where the outcome of the experiment involves a count, the binomial probability distribution is often a better model. 1

2 The binomial distribution can be used when the experiment meets all of the following conditions: 1. There is a fixed, predetermined number of identical trials n. 2. Only two outcomes are possible in each trial, which are success (S) and failure (F). 3. The probability of success, p, is the same in each trial. 4. The trials are independent (the outcome of one has no effect on the others). 5. The random variable x is the number of successes in n trials. 2

3 The shape of the binomial distribution is determined by the values of n and p. We will use the notation b(n, p, x) to represent the value of the binomial distribution with n trials, probability of success p, and x successes. The values are given by the formula: for x = 1, 2, 3,, n 3

4 The distribution will be symmetric if p = 0.5, and skewed when p 0.5. However, even if p 0.5 the distribution becomes approximately symmetric for large values of n. 4

5 Ex 1: Which of the following define x as a binomial random variable. I flip a coin 20 times, and x is the number of heads observed. I flip a coin until a tails is flipped, and x is the number of heads observed. 5

6 Ex 2: Which of the following define x as a binomial random variable. I draw 10 cards at random from a deck of cards (without replacement), and x is the number of face cards drawn. I draw 13 cards at random from a deck of cards (with replacement), and x is the number of hearts drawn. 6

7 NOTE: Replacement is an important issue with the binomial distribution, because in many experiments it allows us to meet Condition 3. However, if the population size is at least 10 times bigger than the sample size, then the difference between replacement or non replacement is small enough to be practically insignificant. In Example 2, the population size (52 cards) is not large enough compared to the sample size (10 cards) for this to be true, so replacement is necessary. 7

8 Finding Probabilities for the Binomial Distribution: Although it is not terribly difficult to calculate b(n, p, x) directly from the formula, we will mostly rely on tables or technology to determine values for the binomial distribution. Table 3 in Appendix A of the text is one such table. The one that I will use is found on my web site (Handouts page), and is also located in my folder in the Q drive. In the first column, locate the appropriate value of n and x. Then locate the correct column to the right based on the value of p. 8

9 Ex 3: Find the probability of observing 7 heads if a fair coin is flipped 10 times. Ex 4: Find the probability of observing at least 7 heads if a fair coin is flipped 10 times. 9

10 Ex 5: A Pew Research study in 2015 found that about 60% of all adults in the U.S. own a smart phone. (a) If 100 American adults were randomly selected, how many of them would you expect to own a smart phone? (b) If 12 American adults are randomly selected, what is the probability that exactly 5 own a smart phone? 10

11 (c) If 12 American adults are randomly selected, what is the probability that no more than 10 own a smart phone? 11

12 Ex 6: Use Minitab to determine the probability for part (b) of Example 5. Open Minitab. Type 5 into the first cell in column C1. Select Calc, Probability Distributions, Binomial. Choose Probability, and enter 12 for the Number of Trials and 0.60 for Event probability. Select Input column and enter C1, then C2 for Optional storage. Click OK. 12

13 Ex 7: Use Minitab to determine the probability for part (c) of Example 5. Open Minitab. Type 10 into the first cell in column C1. Select Calc, Probability Distributions, Binomial. Choose Cumulative probability, and enter 12 for the Number of Trials and 0.60 for Event probability. Select Input column and enter C1, then C2 for Optional storage. Click OK. 13

14 Ex 8: In Example 5, the actual percentage was 64%. Use the formula find the probability that exactly 5 out of 12 randomly selected American adults own a smart phone. 14

15 Center and Spread for the Binomial Distribution: The mean and standard deviation for a binomial distribution can be calculated using some simple formulas: Mean: μ = np Standard deviation: The mean of any probability distribution is referred to as the expected value. Because most outcomes will lie within one standard deviation of the mean, we often say we expect a value of. 15

16 Ex 9: The odds in winning a cash prize in Diamonds Scratchers, a California lottery game, is 1 in Suppose you buy 100 tickets (at $1 each). Consider success to mean that a ticket is a winner (cash prize). (a) Does this experiment meet the conditions for the binomial distribution? Why? (b) Find the mean number of winning tickets. 16

17 (c) Find the standard deviation for the number of winning tickets. (d) How many tickets out of 100 should you expect to be winners? 17

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