To find the probability that at least 1245 Macintosh users will receive the flyer, we can use the binomial probability formula.
The formula for the probability of exactly k successes in n independent Bernoulli trials, each with probability of success p, is:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of exactly k successes
- nCk is the combination formula for selecting k items out of n
- p^k is the probability of k successes
- (1 - p)^(n - k) is the probability of (n - k) failures
In this case, n is the number of computer users that the flyer is being sent to, which is 25,000. The probability of success (p) is the proportion of Macintosh users among all computer users, which is 0.05 (or 5%).
Now, to find the probability that at least 1245 Macintosh users will receive the flyer, we need to calculate the sum of probabilities from 1245 to 25,000.
P(X ≥ 1245) = P(X = 1245) + P(X = 1246) + ... + P(X = 25000).
However, calculating this sum directly would be computationally burdensome. So, an alternative approach is to recognize that the complement of having at least 1245 Macintosh users receive the flyer is having less than 1245 Macintosh users receive the flyer.
To find the probability of the complement event, we can subtract the probability of having less than 1245 Macintosh users from 1.
P(X ≥ 1245) = 1 - P(X < 1245).
P(X < 1245) can be found by calculating the sum of probabilities from 0 to 1244:
P(X < 1245) = P(X = 0) + P(X = 1) + ... + P(X = 1244).
Using the formula mentioned above, we can calculate this probability as well.
Once we have both P(X < 1245) and P(X ≥ 1245), we can subtract the former from 1 to get the probability that at least 1245 Macintosh users will receive the flyer.
Note: Calculating the probabilities directly for such a large number of trials can be computationally intensive. In practice, it is more efficient to use statistical software or programming languages that provide built-in functions to calculate binomial probabilities.