The state bridge design engineer has devised a plan to repair North Carolina's 4420 bridges that are currently listed as being in either poor or fair condition. The state has a total of 13,168 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.)
Binomial distribution
P = 4420/13168 = .33566
1-p = .66434
P(0) = c(5,0) p^0 (1-p)^5
= 1 * 1 * .66434^5 = .129
P(1) = c(5,1)p^1 (1-p)^4
= 5 *.33566 * .66434^4 = .3269123
P(2) = c(5,2)p^2(1-p)^3
= 10 *.33566^2 * .66434^3 = .330347
sum = .657
P(5) = .66434^5 = .12941
To solve these probability questions, we need to know the number of bridges that are rated poor or fair and the total number of bridges in North Carolina.
Let's solve each part of the question one by one:
(a) The probability that the governor will visit no bridges rated as poor or fair:
Since there are 13,168 bridges in total and 4,420 of them are rated as poor or fair, this means that 13,168 - 4,420 = 8,748 bridges are not rated as poor or fair.
To calculate the probability, we divide the number of favorable outcomes (not visiting any poor or fair bridges) by the total number of possible outcomes (visiting any five bridges).
The number of favorable outcomes is selecting all five bridges from the pool of 8,748 bridges:
C(8748, 5) = 8,748 choose 5 = (8,748!)/(5!(8,748 - 5)!)
The total number of possible outcomes is selecting any five bridges from the pool of 13,168 bridges:
C(13168, 5) = 13,168 choose 5 = (13,168!)/(5!(13,168 - 5)!)
Now we can calculate the probability using the formula:
Probability = (favorable outcomes)/(total outcomes)
Probability = C(8748, 5)/C(13168, 5)
Calculate the result using a calculator or software that can handle large factorials.
(b) The probability that the governor will visit one or two bridges rated as poor or fair:
In this case, we need to consider two sets of favorable outcomes:
1. Visiting exactly one bridge rated as poor or fair.
2. Visiting exactly two bridges rated as poor or fair.
We calculate the probabilities for each set separately and then sum them to get the overall probability.
Similar to part (a), calculate the number of favorable outcomes for each set using combinations, and then use the formula:
Probability = (favorable outcomes)/(total outcomes)
(c) The probability that the governor will visit five bridges rated as poor or fair:
Since we want the governor to visit all five bridges rated as poor or fair, the probability will depend on the number of bridges rated as poor or fair.
The probability for this case is:
Probability = (number of bridges rated as poor or fair)/(total number of bridges)
Now you have the framework to calculate the probabilities for each part of the question. Use the appropriate combination formulas and division to get the final answers.