. 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.

(b) [tex]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[/tex]

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

[tex]P(C|E)=\dfrac{P(C\cap E)}{P(E)}=\dfrac{n(C\cap E)}{n(E)}[/tex]

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

[tex]P(C|\sim E)=\dfrac{n(C\cap \sim E)}{n(E)}= \dfrac{150}{500}=\dfrac{3}{10}=0.3[/tex]

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

[tex]P(C\cap E)=P(C)\timesP(E)[/tex] (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,

[tex]P(C)=0.2[/tex], [tex]P(E)=0.5[/tex] (since half the employees exercised)

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

It is seen that

[tex]P(C\cap E)\ne P(C)\timesP(E)[/tex] since

[tex]0.05 \ne 0.2\times0.5[/tex]