Question 939754
  
Given:
a n=10 step Bernoulli experiment (exactly two possible outcomes).
probability is constant at p=38%.
each step of the experiment is independent of the other.
  
Under the given conditions, the problem can be solved using the binomial distribution:
P(X=r)={{{C(n,r)*p^r*(1-p)^(n-r) }}}
where 
r is the number of successes (use credit card because of reward).
n = 10, number of students interviewed in the experiments
p = 0.38, probability of success
C(n,r) = {{{n!/((n-r)!r!)}}}combination of r objects out of n 
  
(a) exactly 2 successes (out of ten)
P(X=2) = {{{C(10,2)*0.38^2*(1-0.38)^8 = 0.1419}}}
Answer: probability of exactly 2 successes is
  
(b) more than 2 successes (i.e. 3 to 10)
It is easier to calculate 1-(P(X=0)+P(X=1)+P(X=2) which gives the same answer.
P(X=0) = {{{C(10,0)*0.38^0*(1-0.38)^10 = 0.0084}}}
P(X=1) = {{{C(10,1)*0.38^1*(1-0.38)^9 = 0.0514}}}
P(X=2) = {{{C(10,2)*0.38^2*(1-0.38)^8 = 0.1419}}}
P(X>2) = 1-(P(X=0)+P(X=1)+P(X=2) = {{{0.0084+0.0514+0.1419}}} = {{{0.2017}}}
Answer: probability of greater than 2 successes is 0.2017
  
(c) between two and five
P(X=2) = {{{C(10,2)*0.38^2*(1-0.38)^8 = 0.1419}}}
P(X=3) = {{{C(10,3)*0.38^3*(1-0.38)^7 = 0.2319}}}
P(X=4) = {{{C(10,4)*0.38^4*(1-0.38)^6 = 0.2487}}}
P(X=5) = {{{C(10,5)*0.38^5*(1-0.38)^5 = 0.1829}}}
P(2<= X <= 5) = P(X=2)+P(X=3)+P(X=4)+P(x=5)={{{0.1419+0.2319+0.2487+0.1829}}}
={{{0.8054}}}
Answer: probability of between 2 to 5 successes is 0.8054