Answer:
To calculate the probability of exactly x = 3 successes in n = 6 trials of a binomial experiment with a probability of success p = 0.4, you can use the binomial probability formula:
P(X = x) = (n choose x) * p^x * (1 - p)^(n - x)
Where:
- n is the number of trials (in this case, n = 6).
- x is the number of successes you want to calculate the probability for (in this case, x = 3).
- p is the probability of success on a single trial (in this case, p = 0.4).
- (n choose x) represents the number of combinations of n items taken x at a time and can be calculated as C(n, x) = n! / (x! * (n - x)!), where "!" denotes factorial.
Now, let's calculate it step by step:
1. Calculate (n choose x):
C(6, 3) = 6! / (3! * (6 - 3)!) = (6 * 5 * 4) / (3 * 2 * 1) = 20
2. Calculate p^x:
p^3 = (0.4)^3 = 0.064
3. Calculate (1 - p)^(n - x):
(1 - p)^(6 - 3) = (1 - 0.4)^3 = (0.6)^3 = 0.216
4. Now, plug these values into the formula:
P(X = 3) = 20 * 0.064 * 0.216
5. Calculate the final result:
P(X = 3) ≈ 0.27648
So, the probability of getting exactly 3 successes in 6 trials with a success probability of 0.4 is approximately 0.27648. The closest option is (c) 0.276.
Explanation: