Explanation:
P(X=4) represents the probability of getting exactly 4 successes in a binomial distribution with n = 12 and p = 0.4.
To calculate P(X=4), we can use the binomial probability formula:
P(X=k) = (nCk) * (p^k) * ((1-p)^(n-k))
Substituting the given values, we have:
P(X=4) = (12C4) * (0.4^4) * ((1-0.4)^(12-4))
Calculating each part:
(12C4) = 12! / (4! * (12-4)!) = 495
(0.4^4) = 0.0256
((1-0.4)^(12-4)) = 0.000137858
Now, multiplying all the values together:
P(X=4) = 495 * 0.0256 * 0.000137858
Rounding to 4 decimal places:
P(X=4) ≈ 0.0069
Next, let's calculate P(X≥4), which represents the probability of getting 4 or more successes.
To calculate P(X≥4), we need to sum the probabilities of getting 4, 5, 6, ..., up to 12 successes.
P(X≥4) = P(X=4) + P(X=5) + P(X=6) + ... + P(X=12)
We have already calculated P(X=4) as 0.0069. We can use the same formula and substitute k with 5, 6, ..., 12 to calculate the remaining probabilities. Then, sum up all the values.
P(X≥4) ≈ P(X=4) + P(X=5) + P(X=6) + ... + P(X=12)
Similarly, let's calculate P(X≤4), which represents the probability of getting 4 or fewer successes.
To calculate P(X≤4), we need to sum the probabilities of getting 4, 3, 2, 1, and 0 successes.
P(X≤4) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
We have already calculated P(X=4) as 0.0069. We can use the same formula and substitute k with 3, 2, 1, and 0 to calculate the remaining probabilities. Then, sum up all the values.
P(X≤4) ≈ P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
Finally, let's calculate P(X>4), which represents the probability of getting more than 4 successes.
To calculate P(X>4), we subtract P(X≤4) from 1.
P(X>4) = 1 - P(X≤4)
Substituting the calculated value of P(X≤4), we have:
P(X>4) ≈ 1 - P(X=4) - P(X=3) - P(X=2) - P(X=1) - P(X=0)
Similarly, P(X<4) represents the probability of getting fewer than 4 successes. To calculate P(X<4), we subtract P(X≥4) from 1.
P(X<4) = 1 - P(X≥4)
Substituting the calculated value of P(X≥4), we have:
P(X<4) ≈ 1 - P(X≥4)
Please note that the approximations in the calculations are due to rounding to 4 decimal places.
Hope this helps.