for n repeated independent trials, with constant probability of success p for all trials, find the probability of exactly x succes n=5,p=1/3, x=4
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To find the probability of exactly x successes in n repeated independent trials with a constant probability of success p for all trials, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Where:
- P(x) is the probability of getting x successes
- C(n, x) is the number of ways to choose x successes from n trials, which can be calculated using the binomial coefficient formula: C(n, x) = n! / (x! * (n-x)!)
- p is the probability of success for each trial
- (1-p) is the probability of failure for each trial
- ^ represents exponentiation
In your case, n = 5, p = 1/3, and x = 4, so we can substitute these values into the binomial probability formula:
P(4) = C(5, 4) * (1/3)^4 * (2/3)^(5-4)
First, let's calculate C(5, 4), which is the number of ways to choose 4 successes from 5 trials:
C(5, 4) = 5! / (4! * (5-4)!)
= 5! / (4! * 1!)
= 5
Now, let's calculate the probability:
P(4) = 5 * (1/3)^4 * (2/3)^1
= 5 * (1/81) * (2/3)
= 10/243
Therefore, the probability of exactly 4 successes in 5 repeated independent trials with a constant probability of success 1/3 is 10/243.