7.5 The Normal Curve Approximation to the Binomial Distribution Advanced Math Topics 7.5 The Normal Curve Approximation to the Binomial Distribution

Find the probability of getting at least 8 heads on 10 flips of a coin. From section 6.5, The Binomial Distribution Formula, we would solve like this… 10C8(.50)8(.50)2 = .0439 10C9(.50)9(.50)1 = .0098 10C10(.50)10(.50)0 = .0010 Add these probabilities up to get 0.0547. The probability of getting 8 or more heads is 5.47%. This took some calculation. We can use the normal curve to help us answer the same question. This is especially helpful when the number of trials becomes large.

μ = np σ =√npq μ = 10(.5) σ =√10(.5)(.5) μ = 5 σ =1.5811 Find the probability of getting at least 8 heads on 10 flips of a coin. Let’s take a look at the probability distribution: The histogram of the probability distribution is shown here: x p(x) .0010 1 .0098 2 .0439 3 .1172 4 .2051 5 .2461 6 7 8 9 10 It is a normal distribution. The question asks for 8 or more heads to be flipped . Thus, we must find the area under the curve to the right of 7.5. Hint: When solving these problems, x is 0.5 more or less toward the outside of the interval you are solving. Find the mean and standard deviation using the formulas in section 6.6. μ = np σ =√npq x - μ 7.5 – 5 μ = 10(.5) σ =√10(.5)(.5) z = = = 1.58 σ 1.5811 μ = 5 σ =1.5811 Find the probability from the mean to a z of 1.58 using the table in the appendix. .5000 - .4429 = .0571 The probability of getting 8 or more heads is 5. 71%.

We are interested in the probability of less A TV company telephones 5,000 people and will cancel a show if less than 1900 viewers are watching the show. Find the probability that the show will be canceled if 40% of viewers watch the show. Steps: 1) Find the mean and standard deviation. μ = 5000(.40) σ = √5000(.40)(.60) μ = np σ = √npq μ = 2000 σ = 34.6410 2) Find out your interval. We are interested in the probability of less than 1900 viewers. (0,1,2….1899 viewers) 3) Add OR subtract 0.5 to the OUTSIDE of your interval. To the outside of the interval is +0.5. Thus, x = 1899.5. x - μ 4) Find your z value. z = x - μ 1899.5 – 2000 σ z = = = -2.90 σ 34.6410 The probability from the mean to 1899.5 is 0.4981. 5) Locate the probability in the Standard Normal curve table. 6) Find the correct probability for the interval. It may help to draw a bell curve. .5000 – 0.4981 = 0.0019 The probability that the show will be canceled is 0.19%.

From the HW P. 385 If 68% of University students have type O blood, what is the probability that a random sample of 700 students will contain 510 or more students with type O blood? .34%

From the HW P. 385 10) 8% of people who borrow from the library do no return the books on time. If 450 people borrowed books from the library today, find the probability that 28 of them will not return the books on time. 2.74%

HW P. 385 #1-12