solving problem the chicago public school system has about 680 schools about 130 of the schools will be closed. if the five schools are chosen at random find the probability that at least three of them willl be closed
You will need to find P(3), P(4), and P(5). Add those values for your probability.
You can use a binomial probability table, or calculate by hand using the following formula:
P(x) = (nCx)(p^x)[q^(n-x)]
x = 3,4,5
n = 5
p = 130/680 = 0.19
q = 1 - p = 1 - 0.19 = 0.81
I'll let you take it from here.
To find the probability that at least three of the five randomly chosen schools will be closed, we need to calculate the number of favorable outcomes and the total number of possible outcomes.
First, let's calculate the number of ways to choose three, four, or five schools out of the 130 that will be closed.
For choosing three schools:
C(130, 3) = (130!)/(3!(130-3)!) = (130!)/(3!127!) = (130*129*128)/(3*2*1) = 5,505,840
For choosing four schools:
C(130, 4) = (130!)/(4!(130-4)!) = (130!)/(4!126!) = (130*129*128*127)/(4*3*2*1) = 20,019,760
For choosing five schools:
C(130, 5) = (130!)/(5!(130-5)!) = (130!)/(5!125!) = (130*129*128*127*126)/(5*4*3*2*1) = 52,566,360
Now, let's calculate the total number of ways to choose any five schools out of the 680 schools in the system:
C(680, 5) = (680!)/(5!(680-5)!) = (680!)/(5!675!) = (680*679*678*677*676)/(5*4*3*2*1) = 257,204,646,916
Now, we can calculate the probability that at least three of the five schools will be closed:
P(at least 3 closed) = P(3 closed) + P(4 closed) + P(5 closed)
P(3 closed) = Number of ways to choose 3 closed schools / Total number of ways to choose 5 schools
P(3 closed) = 5,505,840 / 257,204,646,916
P(4 closed) = Number of ways to choose 4 closed schools / Total number of ways to choose 5 schools
P(4 closed) = 20,019,760 / 257,204,646,916
P(5 closed) = Number of ways to choose 5 closed schools / Total number of ways to choose 5 schools
P(5 closed) = 52,566,360 / 257,204,646,916
Finally, add these probabilities together to get the probability of at least three schools being closed.