In a poll men and women were asked, "when someone yelled or snapped at you at work, how did you want to respond?" Twenty percent of the women in the survey said that they felt like crying. Suppose that this result is true for the current population of women employees. A random sample of 20 women employees is seleceted. Use binomial probabilities table or technology to find the probability that the number of women employees in this sample of 20 who will hold the above opinion in response to the said question is

a. At least 5

b. 1 to 3

c. At most 6

round your answer to four decimal places
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2 Answers

The binomial expansion of (0.2+0.8)^20 provides terms for all possibilities from all 20 women crying to no women crying. The expansion of the binomial expression has the coefficients 1, 20, 190, 1140, ... 20Cr, ... 190, 20, 1. The corresponding probability factors are 0.2^20, 0.2^19*0.8, 0.2^18*0.8^2, ... 0.2^(20-r)0.8^r, ... 0.2^2*0.8^18, 0.2*0.8^19, 0.8^20. These factors are: all 20 with crying response, 19 crying and 1 not crying, 18 crying and 2 not crying, and so on up to only 2 crying, only 1 crying, none crying.

a. At least 5 crying

This includes all women crying, i.e., we need to sum all the binomial terms from r=0 to 15. This comes to 0.3704.

b. 1 to 3

Sum from r=17 to 19 = 0.4000.

c. At most 6

Sum from 14 to 20=0.9133.

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