for n repeated independent trials, with constant probability of success p for all trials, find the probability of exactly x succes n=20,p=1/8, x=2

I know the answer but I do not know how to get to that. Answer is 0.268 should I reduce fraction?

To determine probability of binomial distribution for success (x=2) use the following ...

[n choose x] * p ^x * (1-p) ^ (n-x) where
for the first part ...
[n choose x] =
n! divided by [x! * (n-x!)]
so we have 20! / (2! * 18!)
= 20 * 19 / 2 = 190
the second part we have
p ^ x = (1/8) ^ 2 = 1/64 = 0.015625
the third part we have
(1-p) ^ (n-x) = (7/8) ^ 18 = 0.090395
and finally 190 * 0.015625 * 0.090395 =
0.26836

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 combinations of n items taken x at a time, p^x represents the probability of x successes occurring, and (1-p)^(n-x) represents the probability of (n-x) failures occurring.

In your case, n = 20, p = 1/8, and x = 2. Plugging these values into the formula, we get:

P(2) = (20C2) * ((1/8)^2) * [(1-(1/8))^(20-2)]

To calculate the binomial coefficient (20C2), we use the formula:

(20C2) = 20! / (2! * (20-2)!)

Simplifying further:

P(2) = (20! / (2! * 18!)) * ((1/8)^2) * [(7/8)^18]

Now, we can calculate each part separately:

(20! / (2! * 18!)) = (20 * 19) / (2 * 1) = 190

(1/8)^2 = 1/64

(7/8)^18 = (7^18 / 8^18)

You can either evaluate these calculations individually or simplify the equation by reducing fractions.

Now, substitute the calculated values back into the original equation:

P(2) = 190 * (1/64) * [(7^18) / (8^18)]

Calculating this expression will give you the probability of exactly 2 successes in 20 trials. In this case, the answer is approximately 0.268. If you prefer, you can represent this as a reduced fraction or as a decimal, depending on your preferred format.