PLEASE HELP The Hiking Club plans to go camping in a State park where the probability of rain on any given day is 82%. What is the probability that it will rain on exactly five of the five days they are there? Round your answer to the nearest thousandth.
.82^5 = ?
To find the probability that it will rain on exactly five out of five days, we can use the concept of independent events and the binomial probability formula.
The binomial probability formula is given by:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
Where:
P(X=k) represents the probability of having exactly k successes in n trials.
C(n, k) represents the number of combinations of n items taken k at a time.
p represents the probability of success on each individual trial.
(1-p) represents the probability of failure on each individual trial.
k represents the number of successes we want to find.
n represents the total number of trials.
In this case, we want to find the probability of having exactly five days of rain out of five days, so k = 5, n = 5, and p = 0.82 (probability of rain on any given day).
Let's calculate the probability using the formula:
P(X=5) = C(5, 5) * (0.82^5) * ((1-0.82)^(5-5))
C(5, 5) = 5! / (5! * (5-5)!)
= 1
P(X=5) = 1 * (0.82^5) * ((1-0.82)^(5-5))
= (0.82^5) * (1-0.82)^0
= (0.82^5) * 1
= 0.32768
Therefore, the probability that it will rain on exactly five out of five days is approximately 0.328 (rounded to the nearest thousandth).