A binomial experiment with probability of success =p0.78 and =n7 trials is conducted. What is the probability that the experiment results in fewer than 6 successes?Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)

To solve this problem, we will use the binomial probability formula:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where:

• P(X=k) is the probability of k successes,
• C(n, k) is the combination of n items taken k at a time,
• p is the probability of success,
• n is the number of trials.

We are asked to find the probability of fewer than 6 successes, which means we need to find P(X<6) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5).

Let's calculate each term:

1. P(X=0) = C(7, 0) * (0.78^0) * ((1-0.78)^(7-0))
2. P(X=1) = C(7, 1) * (0.78^1) * ((1-0.78)^(7-1))
3. P(X=2) = C(7, 2) * (0.78^2) * ((1-0.78)^(7-2))
4. P(X=3) = C(7, 3) * (0.78^3) * ((1-0.78)^(7-3))
5. P(X=4) = C(7, 4) * (0.78^4) * ((1-0.78)^(7-4))
6. P(X=5) = C(7, 5) * (0.78^5) * ((1-0.78)^(7-5))

Finally, add all these probabilities together to get the total probability of fewer than 6 successes. Remember to round your final answer to three decimal places.

To solve this problem, we will use the formula for the binomial probability, which is:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where:

• P(X=k) is the probability of k successes in n trials
• C(n, k) is the combination of n items taken k at a time
• p is the probability of success
• n is the number of trials

In this case, we want to find the probability of fewer than 6 successes, which means we need to find the sum of the probabilities for 0, 1, 2, 3, 4, and 5 successes.

So, we calculate each of these probabilities and then add them together:

P(X<6) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)

For each of these, we substitute n=7, p=0.78 into the binomial probability formula and calculate the result.

Finally, we add up all these probabilities to get the final answer. Remember to round your final answer to three decimal places.