Question

(4) (Preview of binomial distribution) Suppose the safe criterion for arsenic in fresh water is less than 340 μg/L. Suppose you want to test the water in 5 wells spread out far enough from each other that a hydrogeologist has confirmed that the 5 wells are drawing from different aquifers (and hence testing should be independent). If the probability of a water sample being clean enough for drinking (i.e., a success) is p: » What is the probability that all 5 tests were successful? » What is the probability that none of the tests were successful? » What is the probability that one of the tests was successful? » What is the probability that two of the tests were successful? . What is the probability that n of the tests were successful, where 0 Sn < 5?

Answer 1

Ans:

Binomial distribution with parameter n=5,probability of success=p

P(x=k)=5C_k*p^k*(1-p)^5-k

k=0,1,2,3,4,5

1)P(all)=P(x=5)=5C₅*p⁵*(1-p)⁰=p⁵

2)P(none)=P(x=0)=5C₀*p⁰*(1-p)^5-0=(1-p)⁵

3)P(x=1)=5C₁*p¹*(1-p)^5-1=5p(1-p)⁴

4)P(x=2)=5C₂*p²*(1-p)³=10p²(1-p)³

5)P(x=n)=5C_n*pⁿ*(1-p)^5-n