Answer:
a) the probability is 0.0734 (7.34%)
b) the probability is 0.7382 (73.82%)
c) 10 boxes
Step-by-step explanation:
Since each box is independent of the others and defining the random variable X= number of boxes of cereal purchased without prices until having 2 prices , then X has a negative binomial distribution . Thus
P(X=k)= C(k+r-1,k)*p^k *(1-p)^r
where
p= probability to not obtain a price when a box is purchased = 1- 0.2 = 0.8
C ( ) = combinations
k = number of boxes without prices
r= number of boxes with prices=2
P(X=k) = probability of purchasing k boxes without prices until obtaining r boxes with prices
a) for k= 8 - r = 6
P(X=6)= C(7,6)*0.8^6*0.2^2 = 7*0.8^6*0.2^2 = 0.0734 (7.34%)
b) for k≥9-2=7
P(X≥7) = 1- P(X<7)= 1- Fn(X=7)
where Fn(X) is the cumulative negative binomial distribution. We can calculate it through its relationship with the cumulative binomial distribution Fb(X) that is easily found in tables :
Fn(k=7,r=2,p=0.8) = Fb(k=7,n=k+r=9,p=0.8) = 0.2618
thus
P(X≥7) = 1- P(X<7)= 1- Fn(X=7) = 1- 0.2618 = 0.7382 (73.82%)
c) the expected value E(X) for a negative binomial distribution is
E(X)= p*r/(1-p) = 0.8*2/0.2 = 8 boxes that do not contain prices
thus
n=k+r = 8 + 2 = 10 boxes
