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Binomial Formula, Mean, and Standard Deviation

Binomial Formula, Mean, and Standard Deviation. Presentation 5.3. A Binomial Situation. Suppose a couple plans to have 3 children. The chance they have a boy is 0.2. The gender of one child is independent of the gender of another child. Let X be the number of boys they have.

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Binomial Formula, Mean, and Standard Deviation

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  1. Binomial Formula, Mean, and Standard Deviation Presentation 5.3

  2. A Binomial Situation • Suppose a couple plans to have 3 children. • The chance they have a boy is 0.2. • The gender of one child is independent of the gender of another child. • Let X be the number of boys they have.

  3. Binomial Specifics • If the chance a child is a boy is 0.2, what’s the chance a child is a girl? • The chance the child is a girl is 0.8. • How many gender sequences (i.e. BBB, BBG, BGG, etc) are possible? • There are 8. • BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG

  4. We want to fill in the probability distribution below: P(X=0) and P(X=3) are already given from the results on the previous page. We only need P(X=1) and P(X=2) now.

  5. Binomial Formula • P(X = 1) = P(only one boy) = P(GGB) + P(GBG) + P(BGG) = [(0.8)2(0.2)] + [(0.8)2(0.2)] + [(0.8)2(0.2)] = 3 (0.8)2(0.2)

  6. Binomial Distribution Similarly, P(X=2) equals 3(0.2)2(0.8). Now we can complete the probability distribution of X.

  7. Binomial Formula • P(X = 1) = P(one boy and two girls) = P(GGB) + P(GBG) + P(BGG) = [(0.8)2(0.2)] + [(0.8)2(0.2)] + [(0.8)2(0.2)] = 3 (0.8)2(0.2) The above form can be generalized into the binomial formula.

  8. Binomial Formula • The formula is: • This is for n trials • The probability of success is p • The probability of failure is 1-p • We are interested in k successes • Then there are n-k failures

  9. Binomial Formula The probability of all the successes happening. The probability of all the failures happening. Number of ways to get success. This is a combination of n choose k. On the TI calculator, this is under Math, Prb and nCr.

  10. Binomial Formula • What is the probability of having all boys when having 3 children? • What is the probability of having 2 boys and 1 girl when having 3 children?

  11. Binomial Formula • What is the probability of having 1 boy and 2 girls when having 3 children? • What is the probability of having all girls when having 3 children?

  12. Example The adult population of a large urban area is 60% black. If a jury of 12 is randomly selected from the adults in this area, what is the probability that precisely 7 jurors are black. Clearly, n=12 and p=.6, so You can confirm this with the calculator by trying: Binompdf (12, .6, 7)

  13. The mean, variance and standard deviation • μ= np • σ 2= np(1-p) • If we have a sample of 20 light bulbs and 5% of all bulbs produced are defective, then on average np=1 bulb will be defective.

  14. Example The AP Statistics Exam contains 40 multiple choice questions each with 5 possible answers. What kind of score might you expect if you randomly guess on each question. Let x be a random variable defined by x = number of correct answers on such an exam Find the mean and standard deviation for x

  15. Example - solution The random variable is clearly binomial with n = 40 and p = 0.2 The mean and standard deviation of x are You should expect to get 8 questions correct

  16. Binomial Formula, Mean, and Standard Deviation • This concludes this presentation.

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