Question

HW 4.4 Score: 13.7/23 11/15 answered Question 12 < > Suppose that 40% of AHC students live in Santa Maria. Find the probability that: Round to three decimals. a. Exactly 7 out of 11 randomly chosen AHC students live in Santa Maria : b. The first AHC student you find that lives in Santa Maria is person number 7 or before : c. More than 7 AHC students out of 11 live in Santa Maria : d. The first Santa Maria resident you find is the 7 th person you ask : Submit Question Jump to Answer

Answer 1

The number of AHC students who live in Santa Maria out of 11 has Binomial distribution with $n=11,p=0.4$ .

The PMF of is $P\left ( X=x \right )=\binom{n}{x}p^x\left ( 1-p \right )^{n-x},x=0,1,2,...,n$ .

a)The required probability is

11 P(X = 7) = 7 P (X = 7) = 0.0701 0.47 (1 – 0.4)11-7

b) Here we have to use Geometric distribution. The PMF is

P(X = x) = (1 - p)*-1p,X = 1,2,3,4,...

The required probability is

$P(X\leqslant 7)=(1-p)^{1-1}p+(1-p)^{2-1}p+...+(1-p)^{7-1}p\\ P(X\leqslant 7)=1-(1-p)^7\\ P(X\leqslant 7)=1-(1-0.4)^7\\ {\color{Blue} P(X\leqslant 7)=0.9720}$

c)Use Binomial distribution.

The required probability is

$P\left ( X>7 \right )=1-P(X\leqslant 7)\\ P\left ( X>7 \right )=1-\sum_{x=0}^{7}\binom{11}{x}0.4^x\left ( 1-0.4 \right )^{11-x}\\ {\color{Blue} P\left ( X>7 \right )=0.0293}$

d) Here we have to use Geometric distribution.

$P(X=7)=(1-0.4)^{7-1}0.4\\ {\color{Blue} P(X=7)=0.0187}$