Macintosh users make up about 5% of all computer users. A computer training school that wants to attract Macintosh users mails an advertising flyer to 25,000 computer users. If the mailing list can be considered a random sample of the population, what is the chance that at least 1245 Macintosh users will receive the flyer?
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.
To find the probability that at least 1245 Macintosh users will receive the flyer, we can use the binomial probability formula.
The probability of success (p) is the proportion of Macintosh users in the population, which is 5% or 0.05. The probability of failure (q) is equal to 1 - p, which is 0.95.
The number of trials (n) is the total number of computer users who receive the flyer, which is 25,000.
Let's calculate the probability using the formula step by step:
Step 1: Calculate the probability of exactly x successes (1245 Macintosh users):
P(X = 1245) = (nCx) * (p^x) * (q^(n-x))
P(X = 1245) = (25000 C 1245) * (0.05^1245) * (0.95^(25000-1245))
Step 2: Calculate the probability of at least x successes (1245 Macintosh users):
P(X >= 1245) = P(X = 1245) + P(X = 1246) + ... + P(X = 25000)
P(X >= 1245) = Σ (P(X = x)), where the sum is from x = 1245 to 25000 inclusive.
Calculating this sum directly might be time-consuming. Instead, we can use the complement rule to find the probability of the opposite event (less than 1245 Macintosh users) and subtract it from 1.
P(X < 1245) = 1 - P(X >= 1245)
Now, we need to calculate the probability of less than 1245 Macintosh users.
P(X < 1245) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 1244)
P(X < 1245) = Σ (P(X = x)), where the sum is from x = 0 to 1244 inclusive.
Using a binomial probability calculator or a statistical software, you can calculate these probabilities more efficiently.