The probability that it will rain on each of the next 5 days is 50%. What is the probability that it will rain for 3 of the next 5 days?
prob = C(5,3) (1/2)^3 (1/2)^2
= 10 (1/32) = 5/16
To find the probability that it will rain for 3 of the next 5 days, we can use the binomial probability formula. The formula is:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
Where:
P(x) is the probability of getting exactly x successes
C(n, x) is the number of ways to choose x successes from n trials, which is calculated as n! / (x! * (n - x)!)
p is the probability of success for each trial
n is the number of trials
In this case, the probability of rain on each day is 50%, so p = 0.5. The number of trials is 5, so n = 5. We want to find the probability of rain for exactly 3 days, so x = 3.
Plugging the values into the formula:
P(3) = C(5, 3) * 0.5^3 * (1 - 0.5)^(5 - 3)
Calculating C(5, 3):
C(5, 3) = 5! / (3! * (5 - 3)!)
= 5! / (3! * 2!)
= (5 * 4 * 3!) / (3! * 2 * 1)
= 5 * 4 / (2 * 1)
= 10
Calculating the probability:
P(3) = 10 * 0.5^3 * (1 - 0.5)^(5 - 3)
= 10 * 0.5^3 * 0.5^2
= 10 * 0.125 * 0.25
= 0.3125
Therefore, the probability that it will rain for 3 of the next 5 days is 0.3125, or 31.25%.