Question

. Last year, at Northern Manufacturing Company, 200 people had colds during the year. One hundred fifty-five people who did no exercising had colds, and the remainder of the people with colds were involved in a weekly exercise program. Half of the 1,000 employees were involved in some type of exercise.(a) What is the probability that an employee will have a cold next year?(b) Given that an employee is involved in an exercise program, what is the probability that he orshe will get a cold next year?(c) What is the probability that an employee who is not involved in an exercise program will get a cold next year?(d) Are exercising and getting a cold independent events? Explain your answer.

Answer 1

Answer:

(a) $\frac{1}{5}$ or 0.2

(b) $\frac{1}{10}$ or 0.1

(c) $\frac{3}{10}$ or 0.3

(d) They are dependent events. See explanation below.

Step-by-step explanation:

Let C = number of employees that had cold = 200

E = number of employees that exercised = 500

(a) $P(C) = \dfrac{200}{1000}=\dfrac{1}{5}=0.2$

(b) $P(C|E)=\dfrac{\text{Number of employees who exercised and had cold}}{\text{Number of employees who exercised}}=\dfrac{50}{500}=\dfrac{1}{10}=0.1$

In probability, the notation $P(C|E)$ means the probability of C given that E has occurred. It is given by

$P(C|E)=\dfrac{P(C\cap E)}{P(E)}=\dfrac{n(C\cap E)}{n(E)}$

(c) $P(C|\sim E)$ denotes the probability of those who did not exercise but had cold. Using the same notations as in (b),

$P(C|\sim E)=\dfrac{n(C\cap \sim E)}{n(E)}= \dfrac{150}{500}=\dfrac{3}{10}=0.3$

(d) When two events, C and E, are independent, then

$P(C\cap E)=P(C)\timesP(E)$ (This is the multiplication law of probability).

It means, in English, the probability of C and E occurring is equal to the product of the probability of C and the probability of E. From the question,

$P(C)=0.2$, $P(E)=0.5$ (since half the employees exercised)

$P(C\cap E)=0.05$ (It is the ratio of the number of those that exercised and had cold to the total number of employees)

It is seen that

$P(C\cap E)\ne P(C)\timesP(E)$ since

$0.05 \ne 0.2\times0.5$