it has been reported that 40% of all internet users have been attacked by viruses. If 20 people are randomly surveyed, find the following probabilities:
a. At least 3 have been attacked
b. No more than 5 have been attacked
c. less than 15 have been attacked
The easiest way to do this problem is to use a binomial distribution table.
For a), n = 20, p = .4, x = 0, 1, 2
Add together the probabilities you find in the table, then subtract from 1 for your answer.
For b), n = 20, p = .4, x = 0, 1, 2, 3, 4, 5
Add together the probabilities you find in the table. This will be your answer.
For c), n = 20, p = .4, x = 15, 16, 17, 18, 19, 20
Add together the probabilities you find in the table, then subtract from 1 for your answer.
I hope this helps.
To find the probabilities, we need to use the binomial distribution formula, which is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes
- C(n, k) is the number of combinations (denoted as n choose k) of selecting k successes from n trials
- p is the probability of success on a single trial
- n is the number of trials
- k is the number of successful outcomes
In this case:
- n = 20 (as we're surveying 20 people)
- p = 0.4 (as 40% of internet users have been attacked)
a. At least 3 have been attacked:
To find this probability, we need to calculate the probability of getting exactly 3, 4, 5, ..., 20 people attacked and sum them up:
P(at least 3 attacked) = P(3) + P(4) + P(5) + ... + P(20)
Calculating each probability separately would be tedious, so it's easier to calculate the complement and subtract it from 1:
P(at least 3 attacked) = 1 - P(0) - P(1) - P(2)
b. No more than 5 have been attacked:
To find this probability, we need to calculate the probability of getting 0, 1, 2, 3, 4, or 5 people attacked and sum them up:
P(no more than 5 attacked) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5)
c. Less than 15 have been attacked:
To find this probability, we need to calculate the probability of getting 0, 1, 2, ..., 14 people attacked and sum them up:
P(less than 15 attacked) = P(0) + P(1) + P(2) + ... + P(14)
Now, let's calculate these probabilities using the formula.