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Binomial Probability Distribution Questions

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1. A fair coin is flipped six times. What is:
a. The probability of zero heads
b. The probability of 1 heads?
c. The probability of 4 heads?
d. The expected value of the number of heads? (Show a calculation to verify your answer.)
e. The standard deviation of the number of heads? (Show a calculation to verify your answer.)

2. A research team at Cornell University conducted a study that found that 10% of businessmen wear a tie so tight that it reduces blood flow to the brain (source: Changes: Risks and Odds in Everyday Life, by James Burke). At a board meeting of 20 businessmen, what is:
a. The probability that three men are wearing a tie that is too tight?
b. The probability that less than five men are wearing a tie that is too tight?
c. The expected number of men who are wearing a tie that is too tight? (Show a calculation to verify your answer.
d. The standard deviation of the number of men who are wearing a tie that is too tight? (Show a calculation to verify your answer.)

3. The percentage of men in a heterosexual marriage who say that they would marry the same woman again if they had to do it all over again is 80% (Source: Harper's Index). If we randomly sampled 25 men in heterosexual marriages, what is:
a. The probability that all 25 would say that they would marry the same woman again if they had to do it all over?
b. The probability that 20 would say that they would marry the same woman again if they had to do it over?
c. The probability that fewer than 5 men would say that they would marry the same woman again if they had to do it over?
d. The mean (i.e. the expected number of men who would say that they would marry the same woman again if they had to do it over? (Show a calculation to verify your answer.)
e. The standard deviation of the number of men who would say that they would marry the same woman again if they had to do it over? (Show a calculation to verify your answer.)

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Solution Summary

Calculations are given for each problem. The relative frequency histogram is provided in the pdf file attached.

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1a
Half the time, the coin is expected to land heads, and half the time, the coin is expected to land tails.
1 - 0.5 = 0.5
C(6,0) * 0.5^0 * 0.5^6 = 0.015625

b
C(6,1) * 0.5^1 * 0.5 ^ 5 = 0.09375

c
C(6,4) * 0.5^4 * 0.5^2 = 0.234375

d
6 * 0.5 = 3

e
sqrt ( 6 * 0.5 * 0.5) = sqrt (1.5) = 1.2

2a
1 - 0.1 = 0.9
C(20,3) * 0.1^3 * 0.9^17 = 0.1901

b
P( 0 men are wearing a tie that is too tight) = C(20,0) * 0.1^0 * 0.9^20 = 0.1216
P( 1 man is wearing a tie that is too tight) = C(20,1) * 0.1^1 * 0.9^19 = 0.2702
P( 2 ...

Solution provided by:
J Pye, MSc
Education
  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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