Researchers often enter a lot of data into statistical software programs. The probability of making zero to two errors per 1,000 keystrokes is 0.41, and the probability of making three to five errors per 1,000 keystrokes is 0.22. Find the probabilities (per 1,000 keystrokes) associated with each of the following.(a) at most two errors(b) at least three errors(c) at most five errors(d) more than five errors

(a) P(0≤x≤2) = 0.41

(b) P(x≥3) = 0.59

(c) P(x≤5) = 0.63

(d) P(x≥6) = 0.37

Step-by-step explanation:

(a) The probability to have at most two errors is the probability to have 0, 1 or 2 errors or the probability of making zero to two errors. So, the probability to have at most two error is:

P(0≤x≤2) = 0.41

(b) The probability to have at least three errors is the probability to have 3 or more errors. So, it can be calculated as:

P(x≥3) = 1 - P(x≤2)

P(x≥3) = 1 - 0.41

P(x≥3) = 0.59

(c) The probability to have at most five error is the probability to have 0, 1, 2, 3, 4 or 5 errors. This can be calculated as the sum of the probability to have zero to two errors and the probability to have three to five errors as:

P(x≤5) = P(0≤x≤2) + P(3≤x≤5)

P(x≤5) = 0.41 + 0.22

P(x≤5) = 0.63

(d) The probability to have more than five errors is the probability to have 6 or more errors. So, it can be calculated as:

P(x≥6) = 1 - P(x≤5)

P(x≥6) = 1 - 0.63

P(x≥6) = 0.37