For n repeated independent trials, with constant probability of success p for all trials, find the probability of exactly x successes. Round your answer to four decimal places
n = 10, p = 1/7, x = 3
A. 0.1189
B. 0.1546
C. 0.1308
D. 0.1427
B
.1546
To find the probability of exactly x successes in n repeated independent trials with a constant probability of success p, we can use the binomial probability formula.
The formula is given by:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
Where nCx represents the number of combinations of n items taken x at a time, p^x represents the probability of x successes, and (1-p)^(n-x) represents the probability of (n-x) failures.
Using the given values n = 10, p = 1/7, and x = 3, we can plug them into the formula:
P(3) = (10C3) * ((1/7)^3) * ((1 - 1/7)^(10 - 3))
Using the combination formula (nCx) = n! / (x!(n-x)!)
P(3) = (10! / (3!(10-3)!)) * ((1/7)^3) * ((6/7)^7)
Calculating the combination:
P(3) = (10! / (3!7!)) * (1/343) * (279936/823543)
P(3) = (10 * 9 * 8) / (3 * 2 * 1) * (1/343) * (279936/823543)
P(3) = 720 / 343 * 279936/823543
P(3) = 0.1189 (rounded to four decimal places)
Therefore, the probability of exactly 3 successes in 10 repeated independent trials with a constant probability of success 1/7 is A) 0.1189.
To find the probability of exactly x successes in n repeated independent trials with a constant probability of success p, you can use the binomial probability formula:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where nCx represents the number of ways to choose x successes from n trials, p^x represents the probability of x successes, and (1-p)^(n-x) represents the probability of (n-x) failures.
In this case, n = 10, p = 1/7, and x = 3. Now we can substitute these values into the formula and calculate the probability.
P(3) = (10C3) * (1/7)^3 * (1 - 1/7)^(10-3)
To calculate (10C3), which represents the number of ways to choose 3 successes from 10 trials, you can use the combination formula:
(10C3) = 10! / (3! * (10-3)!)
Plugging in the values:
(10C3) = 10! / (3! * 7!)
Calculating the factorials:
(10C3) = 10 * 9 * 8 / (3 * 2 * 1) = 120
Now we can substitute all the values into the original formula:
P(3) = 120 * (1/7)^3 * (1 - 1/7)^(10-3)
P(3) ≈ 0.1308
Therefore, the probability of exactly 3 successes in 10 repeated independent trials with a constant probability of success 1/7 is approximately 0.1308.
So, the correct answer is C. 0.1308.
I believe I answered the question, just find the probability of
7 failures ( a 6/7 chance each time)
3 successes (a 1/7 chance each time)
Then multiply.