Question

There are 750 players on the active rosters of the 30 major league baseball teams. A random sample of 16 players is to be selected and tested for use of illegal drugs. (Give your answers correct to three decimal places.)

(a) If 4% of all the players are using illegal drugs at the time of the test, what is the probability that 1 or more players test positive and fail the test?
  

(b) If 19% of all the players are using illegal drugs at the time of the test, what is the probability that 1 or more players test positive and fail the test?
  

(c) If 29% of all the players are using illegal drugs at the time of the test, what is the probability that 1 or more players test positive and fail the test?

  

The state bridge design engineer has devised a plan to repair North Carolina's 4140 bridges that are currently listed as being in either poor or fair condition. The state has a total of 13,420 bridges. Before the governor will include the cost of this plan in his budget, he has decided to personally visit and inspect five bridges that are to be randomly selected.

(a) What is the probability that the governor will visit no bridges rated as poor or fair. (Give your answer correct to three decimal places.)   

(b) What is the probability that the governor will visit one or two bridges rated as poor or fair. (Give your answer correct to three decimal places.)


(c) What is the probability that the governor will visit five bridges rated as poor or fair. (Give your answers correct to five decimal places.)

Answer 1

n = 16
As per binomial distribution,
P(X=r) = nCr * p^r * (1-p)^(n-r)

a)
p = 0.04

P(x > =1) = 1 - P(x < =0)

= 1- 16C0 * 0. 04^0 * 0.96^16
= 1 - 0.520
= 0.480

b)

p = 0.19

P(x > =1) = 1 - P(x < =0)

= 1- 16C0 * 0. 19^0 * 0.81^16
= 1 - 0.034
= 0.966

c)

p = 0.29

P(x > =1) = 1 - P(x < =0)

= 1- 16C0 * 0.29^0 * 0.71^16
= 1 -0.004
= 0.996

2)

Here, p = 4140 /13420 = 0.31 , n = 5

a)

P(x = 0) = 5C0 * 0.31^0 * 0.69^5
= 0.156

b)

P(x > 1| 2) = P(x > 1) + P(x > 2)

= 1 - P(x < =1) + 1 - P(x < =2)
= 1 - P(x =0 ) - P(x =1) + 1 - P(x =0) - P(x =1) - P(x =2)
= 1 - 5C0 * 0.31^0 * 0.69^5 - 5C1* 0.31^1 * 0.69^4 + 1 - 5C0 * 0.31^0 * 0.69^5 - 5C1 * 0.31^1 * 0.69^4 -5C2 * 0.31^2 * 0.69^3

= 0.492 + 0.177 = 0.669

c)

P(x = 5) = 5C5 * 0.31^5 * 0.69^0
= 0.00286