63 % of men consider themselves professional baseball fans. You randomly select 10 men and ask each if he considers himself a professional baseball fan. Find the probability that the number who consider themselves baseball fans is (a) exactly five, (b) at least six, and (c) less than four.

a. 0.00069

b. 0.063

c. 0.00095

Step-by-step explanation:

a. Probability that exactly 5 men consider themselves professional baseball fans:-

Here, there's a 63/100 or 63% possiblity that five (5) men consider themselves fans and of course, a 37/100 possibility that the other 5 men consider themselves non fans. Each of these must be multiplied and then the result will be the probability that exactly five men consider themselves baseball fans:-

(63/100) ^5 * (37/100) ^5

= 0.09924 * 0.006934

= 0.000688

Therefore the probability that 5 consider themselves baseball fans is 0.00069

b. The probability that at least 6 are fans:-

In this case, each of the 6 men have some 63/100 or 63% possibility of being a fan.

We therefore multiply 63/100 by itself up to six times

i. e 63/100 * 63/100 * 63/100 * 63/100 * 63/100 * 63/100

or (63/100) ^6

= 0.06252

The probability that at least 6 men consider themselves professional baseball fans is 0.063

c. Probability that less than four consider themselves baseball fans:-

In this scenario, it means that at least 7 men out of the ten do not consider themselves professional baseball fans. We then simply multiply 37/100 by itself up to seven times. It isn't too necessary what the remaining three are.

= (37/100) ^7

= 0.0009493

Therefore the possibility that less than four men do not consider themselves baseball fans is 0.00095