The binomial distribution can be used to model the given situation because:
1. The experiment is Bernoulli, i.e. with exactly two outcomes (because of rewards program or not)
2. Probability is known (29%) and remains constant throughout the experiment
3. The number of trials is known and not variable (10)
4. Sampling is assumed random (given) and independent(assumed from context).
The probability of selecting x successes (because of rewards program) from a sample of size n=10 with probability p (=0.29) is given by the binomial distribution as:
P(x,n,p)=C(n,x)*(p^x)*(1-p)^(n-x)
where C(n,x) is the number of combinations of x object from n, given by C(n,x)=n!/(x!(n-x)!)
(a) Probability that exactly two students say they use credit cards because of the rewards program
P(X=2)
=P(2,10,0.29)
=C(10,2)*(0.29^2)(0.71^8)
=0.244
(b) Probability that MORE than two students say they use credit cards because of the rewards program
P(X>2)
=1-P(X<=2)
=1-(P(X=0)+P(X=1)+P(X=2))
=1-P(0,10,0.29)+P(1,10,0.29)+P(2,10,0.29))
=1-(C(10,0)*(0.29^0)(0.71^10)+C(10,1)*(0.29^1)(0.71^9)+C(10,2)*(0.29^2)(0.71^8))
=1-(0.033+0.133+0.244)
=1-0.410
=0.590
