Final answer:
The probability P(6) of getting exactly 6 successes in a binomial experiment with 9 trials and a success probability of 0.3 is calculated using the binomial formula P(X = x) = nCx * p^x * q^(n-x), with q = 1 - p, and the binomial coefficient 9C6.
Step-by-step explanation:
Binomial Probability Calculation
To determine the probability P(6) of obtaining exactly 6 successes in a binomial experiment with 9 trials (n = 9) and a success probability of p = 0.3, we use the binomial probability formula which is:
P(X = x) = nCx * p^x * q^(n-x)
Where:
X is the random variable representing the number of successes,
n is the number of trials,
x is the specific number of successes for which we are calculating the probability (in this case, x = 6),
p is the probability of success in a single trial,
q is the probability of failure (q = 1 - p).
So for our problem:
q = 1 - 0.3 = 0.7
Now, the binomial coefficient nCx (in this case, 9C6) is the number of ways to choose x successes from n trials, which can be calculated as n!/(x!(n-x)!).
Therefore, the probability of getting exactly 6 successes out of 9 trials where the success probability is 0.3 would be calculated as:
P(6) = 9C6 * (0.3)^6 * (0.7)^3
Plugging the numbers into a calculator, we would get the exact probability value for P(6).