64% 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.

Here the situation satisfies the following criteria for binomial distribution:
1. Bernoulli trials, i.e. exactly two possible outcomes (professional or not)
2. Number of trials is known before and constant throughout the experiment, i.e. independent of outcomes (10 trials)3. All trials are independent of each other (assumed from context)4. Probability of success is known, and remain constant throughout trials. (success=claims to be professional, p=0.64)Since all criteria are satisfied, we can model with binomial distribution, where the probability of x successes out of N trials each with probability of success p is given byP(x)=C(N,x)(p^x)(1-p)^(N-x)and,C(N,x) is number of combinations of selecting x objects out of N.

(a) exactly 5, so
P(5)=C(10,5)(0.64)^5*(0.36)^5
=0.1636

(b) P(x>=6)
=P(x=6)+P(x=7)...+P(x=10)
=0.24239+0.24623+0.16416+0.06485+0.01153
=0.72916

(c) P(x<4)=P(x=0)+P(x=1)+P(x=2)+P(x=3)
=0.00004+0.00065+0.00520+0.02465
=0.03054