1. A questionnaire asks shareholders of a company to state whether they consider the
chairman’s salary to be too high, about right, or too low. Excluding shareholders who
have no opinion, the probabilities of answers from randomly selected shareholders are as
follows:
Too High 0.82
About Right 0.12
Too Low 0.05
What are the probabilities that if three shareholders are selected at random;
a. They will all answer ‘too high’
b. Exactly two will answer ‘too high’
c. Exactly two will give the same answer

A common discrete distribution is used in statistics, as opposed to a continuous distribution is called a Binomial distribution. It is given by the formula,

P(x) = ⁿCₓ (pˣ) (q⁽ⁿ⁻ˣ⁾)

Where,

x is the number of successes needed,

n is the number of trials or sample size,

p is the probability of a single success, and

q is the probability of a single failure.

A.) The probability of a shareholder answering too high is 0.82. Therefore, the probability that if three shareholders are selected at random they will all answer ‘too high’ is,

P(x) = ⁿCₓ (pˣ) (q⁽ⁿ⁻ˣ⁾)

P(x=3) = ³C₃ (0.82)³ (1-0.82)⁰

P(x=3) = 0.551368

Hence, the probability that if three shareholders are selected at random they will all answer ‘too high’ is 0.5514.

B.) The probability of a shareholder answering too high is 0.82. Therefore, the probability that if three shareholders are selected at random Exactly two will answer ‘too high’ is,

P(x) = ⁿCₓ (pˣ) (q⁽ⁿ⁻ˣ⁾)

P(x=2) = ³C₂ (0.82)² (1-0.82)¹

P(x=2) = 0.363096

Hence, the probability that if three shareholders are selected at random Exactly two will answer ‘too high’ is 0.363096.

C.) The probability of a shareholder answering too high is 0.82. Therefore, the probability that if three shareholders are selected at random Exactly two will give the same answer is,

P = ³C₂ (0.82)² (1-0.82)¹ + ³C₂ (0.12)² (1-0.12)¹ + ³C₂ (0.05)² (1-0.05)¹

P = 0.363096 + 0.038016 + 0.007125

P = 0.408237

Hence, the probability that if three shareholders are selected at random Exactly two will give the same answer is 0.408237.

Learn more about Binomial Distribution:

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