The probability that a person catches a cold during the cold and flu season is 0.62. Assume that 5 people are chosen at random.
A) What is the probability that exactly four of them will catch a cold?
B) What is the probability that three or more of them will catch a cold?
C) On average, how many of these five people would you expect to catch a cold?
prob of cold = .62
prob of not cold = .38
a) prob of 4 of 5 catch cold
= C(5,4) (.62)^4 (.38)
b) prob 3 or more
= prob 3 + prob 4 + prob 5
= C(5,3)(.62)^3 (.38)^2 + C(5,4) (.62)^4 (.38) + C(5,5) .62^5
= ...
To find the probabilities in this scenario, we can use the binomial probability formula: P(x) = C(n, x) * p^x * (1-p)^(n-x), where P(x) is the probability of x successes, n is the total number of trials, p is the probability of success in each trial, and C(n, x) is the number of combinations of n items taken x at a time.
Let's calculate the probabilities one by one:
A) To find the probability that exactly four of them will catch a cold, we need to find P(4). Since there are 5 people chosen at random, n = 5. The probability of catching a cold is 0.62, so p = 0.62. To calculate P(4):
P(4) = C(5, 4) * 0.62^4 * (1-0.62)^(5-4)
Using the combination formula: C(5, 4) = 5! / (4!(5-4)!) = 5
P(4) = 5 * 0.62^4 * (1-0.62)^(5-4)
Calculating the values:
P(4) = 5 * 0.62^4 * (0.38)^1 = 5 * 0.062416 * 0.38 = 0.11871504
So, the probability that exactly four of them will catch a cold is approximately 0.1187.
B) To find the probability that three or more of them will catch a cold, we need to find P(3) + P(4) + P(5). We have already calculated P(4) in part A. Now, let's calculate P(3):
P(3) = C(5, 3) * 0.62^3 * (1-0.62)^(5-3)
Using the combination formula: C(5, 3) = 5! / (3!(5-3)!) = 10
P(3) = 10 * 0.62^3 * (1-0.62)^(5-3)
Calculating the values:
P(3) = 10 * 0.238328 * 0.38 = 0.9041624
Now, adding P(3), P(4), and P(5), we get:
P(3 or more) = P(3) + P(4) + P(5) = 0.9041624 + 0.11871504 + P(5)
To find P(5), we know that there are only two possibilities: exactly 4 or exactly 5 people catching a cold. So, P(5) = 1 - (P(3) + P(4)):
P(5) = 1 - (0.9041624 + 0.11871504) = 1 - 1.02287744 = -0.02287744
Since probabilities cannot be negative, the probability of exactly 5 people catching a cold is not possible. Therefore, P(5) = 0.
Therefore:
P(3 or more) = 0.9041624 + 0.11871504 + 0 = 1.02287744
So, the probability that three or more of them will catch a cold is approximately 1.
C) To find the average number of people who would catch a cold, we can multiply the number of people (5) by the probability of catching a cold (0.62).
Average = 5 * 0.62 = 3.1
Therefore, on average, we would expect approximately 3.1 people out of five to catch a cold.