Answer to Question #245742 in Statistics and Probability for clem

Question

When people visit a certain large shop, on average 34% of them do not buy anything, 53% spend less 

than $50 and 13% spend at least $50.

(i) 15 people visiting the shop are chosen at random. Calculate the probability that at least 14 of 

them buy something.

(ii) n people visiting the shop are chosen at random. The probability that none of them spends at 

least $50 is less than 0.04. Find the smallest possible value of n.

Answer 1

Solution:

$n=15,q=0.34,p=0.66$

(i) $P(X\ge 14)=P(X=14)+P(X=15)$

$=^{15}C_{14}(0.66)^{14}(0.34)^1+^{15}C_{15}(0.66)^{15}(0.34)^0 \\=0.0171$

(ii) $p=0.13,q=0.87$

$P(X=0)<0.04$

$\Rightarrow ^nC_0(0.87)^n(0.13)^0<0.04$

$\Rightarrow (0.87)^n<0.04$

Using hit and trial, put n=23

$LHS=(0.87)^{23}=0.041>0.04$

Put n=24.

$LHS=(0.87)^{24}=0.035<0.04$

Thus, required value of n=24.