Questions: A binomial experiment with probability of success p=0.6 and n=6 trials is conducted. What is the probability that the experiment results in exactly 2 successes? Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)

A binomial experiment with probability of success p=0.6 and n=6 trials is conducted. What is the probability that the experiment results in exactly 2 successes?

Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)

Transcript text: A binomial experiment with probability of success $p=0.6$ and $n=6$ trials is conducted. What is the probability that the experiment results in exactly 2 successes? Do not round your intermedlate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)

Solution

Solution Steps

Step 1: Calculate the binomial coefficient

The binomial coefficient $\binom{n}{k}$ is calculated as $\frac{n!}{k!(n-k)!}$, which is $\frac{720!}{2!(24!}$ = 15.

Step 2: Calculate the probability of k successes and n-k failures

The probability of exactly 2 successes ($p^k$) is calculated as $0.6^{2} = 0.36$, and the probability of 4 failures ($(1-p)^{n-k}$) is $(1-0.6)^{4} = 0.0256$.

Step 3: Calculate the final probability

The final probability $P(X = k)$ is calculated as the product of the binomial coefficient, the probability of 2 successes, and the probability of 4 failures, which is 15 * 0.36 * 0.0256 = 0.138.

Final Answer:

The probability of achieving exactly 2 successes in 6 trials of a binomial experiment, where the probability of success in each trial is 0.6, is approximately 0.138.

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