Question

a fair die is rolled 3 times. A 4 is considered "success", while all other outcomes are "failures." find the probability of the number of success
a) 2
b)3

Answer 1

Use binomial distribution, which requires a known and constant probability of success, a known number of trials, independence of trials.
probability of success, p=1/6
number of trials, n=3
the probability of x successes is
P(x)=C(n,x)p^x (1-p)^(n-x)
where
C(n,x) is number of combinations choosing x objects out of n, and
C(n,x)=n!/(x!(n-x)!)

(a) x=2 successes
P(2)=C(3,2)(1/6)^2 (5/6)^1
=(6/(2*1))*(1/6)^2*(5/6)
=5/72

(b) x=3 successes
P(3)=C(3,3)(1/6)^3(5/6)^0
=(1)(1/216)(1)
=1/216