(A) To find the binomial probability P(x = 4), where n = 12 and p = 0.30, we can use the formula for binomial probability:
P(x) = (nCx) * p^x * (1-p)^(n-x)
where nCx represents "n choose x", and it can be calculated using the formula:
nCx = n! / (x! * (n-x)!)
In this case, n = 12, x = 4, and p = 0.30. Let's calculate P(x = 4):
P(4) = (12C4) * (0.30^4) * (0.70^8)
Calculating the values:
12C4 = 12! / (4! * (12-4)!) = 495
P(4) = 495 * (0.30^4) * (0.70^8) = 0.152
So, P(x = 4) is approximately 0.152.
(B) To set up the binomial probability P(x is at most 4) using probability notation, we can write:
P(x ≤ 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4)
This represents the sum of the probabilities of all possible outcomes from x = 0 to x = 4.
(C) To find the normal approximation to the binomial probability P(x = 4) in part A, we can use the formula for the mean (µ) and standard deviation (σ) of a binomial distribution, which are given by:
µ = n * p
σ = √(n * p * (1-p))
In this case, n = 12 and p = 0.30. Let's calculate µ and σ:
µ = 12 * 0.30 = 3.6
σ = √(12 * 0.30 * (1-0.30)) ≈ 1.799
Now, we can use these values to find the normal approximation for P(x = 4) using the formula for the normal distribution:
P(x = 4) ≈ P(3.5 < x < 4.5)
So, the final formula for the normal approximation to the binomial probability P(x = 4) is:
P(3.5 < x < 4.5) = Φ((4.5 - µ) / σ) - Φ((3.5 - µ) / σ)
where Φ represents the cumulative distribution function of the standard normal distribution.