a machine has 7 identical components which function independently. the probability that a component will fail is 0.2. the machine will stop working if more than three components fail. find the probability that the machine will be working.

To find the probability that the machine will be working, we need to find the probability that three or fewer components fail.

Let's devise a solution strategy:

1. Calculate the probability of a single component failing.
2. Use this probability to find the probability of three or fewer components failing.
3. Subtract this probability from 1 to find the probability that the machine will be working.

Let's begin by calculating the probability of a single component failing. We are given that the probability of a component failing is 0.2. Therefore, the probability of a component working is 1 - 0.2 = 0.8.

Now, let's calculate the probability of three or fewer components failing. Since the components function independently, we can use the binomial probability formula:

P(X ≤ k) = ∑ (n choose k) * p^k * (1 - p)^(n - k)

Where:
- P(X ≤ k) is the probability of X (number of components failing) being less than or equal to k.
- (n choose k) is the binomial coefficient. It represents the number of ways to choose k components out of n.
- p is the probability of a single component failing.
- (1 - p) is the probability of a single component working.
- n is the total number of components.

Using this formula, we can calculate the probability of three or fewer components failing:

P(X ≤ 3) = ∑ (7 choose k) * 0.2^k * 0.8^(7 - k), where k = 0, 1, 2, 3

Let's calculate it step by step:

P(X = 0) = (7 choose 0) * 0.2^0 * 0.8^7 = 1 * 1 * 0.8^7
P(X = 1) = (7 choose 1) * 0.2^1 * 0.8^6 = 7 * 0.2 * 0.8^6
P(X = 2) = (7 choose 2) * 0.2^2 * 0.8^5 = 21 * 0.04 * 0.8^5
P(X = 3) = (7 choose 3) * 0.2^3 * 0.8^4 = 35 * 0.008 * 0.8^4

Finally, we can find the probability that the machine will be working:

P(machine working) = 1 - P(X ≤ 3)

Calculate the values above and subtract their sum from 1 to obtain the final answer.