1. For n repeated independent trials, with constant probability of success p for all trials, find the probability of exactly x successes in the following exercises.Round your final answers to three decimal places.
a.n = 7, p = ¼ , x = 4
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 binomial probability formula is given by:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- P(x) is the probability of exactly x successes
- n is the number of trials
- p is the probability of success in a single trial
- nCx is the number of combinations of n items taken x at a time, which can be calculated using the formula nCx = n! / (x!(n-x)!), where '!' denotes the factorial operation (e.g., 5! = 5 * 4 * 3 * 2 * 1)
Let's solve the exercise using the given values:
n = 7, p = 1/4, x = 4
P(4) = (7C4) * (1/4)^4 * (1 - 1/4)^(7-4)
First, let's calculate the combinations part:
7C4 = 7! / (4!(7-4)!)
= (7 * 6 * 5 * 4!) / (4! * 3!)
The 4! in the numerator and denominator cancels out, leaving:
7C4 = (7 * 6 * 5) / (3 * 2 * 1)
= 35
Next, substitute the values into the binomial probability formula:
P(4) = 35 * (1/4)^4 * (3/4)^(7-4)
Now, evaluate the exponentials:
(1/4)^4 = 1/256
(3/4)^(7-4) = (3/4)^3 = 27/64
Finally, calculate the probability P(4):
P(4) = 35 * 1/256 * 27/64
≈ 0.067
Therefore, the probability of exactly 4 successes in 7 repeated independent trials with a success probability of 1/4 is approximately 0.067, rounded to three decimal places.