P(X≥40)≈1−∑ k=039​ ( k200​ )×(0.28) k ×(0.72) 200−k

The given expression is a probability calculation using the binomial distribution. Here's how you can solve it step by step:

1. Identify the parameters: In this case, the number of trials is 200 (n=200), the probability of success on each trial is 0.28 (p=0.28), and we want to find the probability that the number of successes is greater than or equal to 40 (X≥40).

2. Use the complement rule: The complement rule states that the probability of an event happening is 1 minus the probability of it not happening. In this case, P(X≥40) is the complement of P(X<40), so we can write P(X≥40) = 1 - P(X<39).

3. Calculate P(X<39): This is the sum of the probabilities of getting k successes for each k from 0 to 39. This can be calculated using the binomial probability formula: P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k)), where C(n, k) is the binomial coefficient "n choose k".

4. Sum up the probabilities: Add up the calculated probabilities from step 3 for all k from 0 to 39.

5. Subtract from 1: Finally, subtract the sum from step 4 from 1 to get P(X≥40).

Note: This calculation can be quite complex and time-consuming to do by hand, especially with large numbers like 200 trials. It's typically done using statistical software or a scientific calculator that can handle binomial distributions.