Question 1204358: Trees planted by a landscaping firm have a 95% one-year survival rate, If they plant 15 trees in a park, what is the following probabilities:
1. All the trees survive one year.
Answer:
2. At least 13 trees survive one year.
Answer:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i think you are looking at a binomial probability type of problem.
if so, then:
p = .95 = probability that a tree survives for one year.
q = .05 = probability that a tree doesn't survive one year.
binomial probability formula is:
p(x) = p^x * q^(n-x) * c(n,x).
c(n,x) is the number of possible combinations of n elements taken x at a time.
c(n,x) is equal to n! / (x! * (n-x)!).
for example, c(15,10) = 15! / (10! * 5!) = (15 * 14 * 13 * 12 * 11 * 10!) / (10! * 5!) which is equal to (15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1) which is equal to 3003.
you can see that in the spreadsheet on the row where x = 5.
the probability of 0 trees surviving for one year is equal to .95^0 * .05^15 * c(15,0) = 3.05175781 * 10^-20.
the probability of at least 1 tree surviving for one year is equal to 1 minus that = 1.
the probaiblity that at least 13 trees survive for one year is equal to p(13) + p(14) + p(15).
p(13) = .95^13 * .05^2 * c(15,13) = .1347522969.
p(14) = .95^14 * .05^1 * c(15,14) = .3657562343.
p(15) = .95^15 * .05^0 * c(15,15) = .4632912302.
p(13) + p(14) + p(15) = .9637997614.
all the probabilities are shown in the following excel spreadsheet.
in that spreadsheet, p = .95, q = .05, n = 15, x = 0 to 15.
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