Question

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

Answer 1

Using the binomial distribution, it is found that the probabilities are given as follows:

a) 0.5514 = 55.14%.

b) 0.3631 = 36.31%.

c) 0.4082 = 40.82%.

What is the binomial distribution formula?

The formula is:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• n is the number of trials.

• p is the probability of a success on a single trial.

Item a:

In this problem, we have p = 0.82, n = 3, and the probability is P(X = 3), hence:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 3) = C_{3,3}.(0.82)^{3}.(0.18)^{0} = 0.5514$

Item b:

In this problem, we have p = 0.82, n = 3, and the probability is P(X = 2), hence:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 2) = C_{3,2}.(0.82)^{2}.(0.18)^{1} = 0.3631$

Item c:

We have to find P(X = 2) for the three probabilities, p = 0.82, p = 0.12, p = 0.05 and add them, hence:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 2) = C_{3,2}.(0.82)^{2}.(0.18)^{1} = 0.3631$

$P(X = 2) = C_{3,2}.(0.12)^{2}.(0.88)^{1} = 0.0380$

$P(X = 2) = C_{3,2}.(0.05)^{2}.(0.95)^{1} = 0.0071$

Then:

p = 0.3631 + 0.0380 + 0.0071 = 0.4082.

More can be learned about the binomial distribution at https://brainly.com/question/24863377

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