1. A door-to-door salesperson has found that her success rate in selling an item to a customer is 0.2. If the salesperson contacts 20 persons, what is the probability that:
a. She sells to all three?
b. She sells to exactly one?
c. She sells to no one?
To find the probabilities, we need to use the binomial probability formula. The formula is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting k successes out of n trials
C(n, k) is the number of combinations of n items taken k at a time (n choose k)
p is the probability of success in one trial
(1-p) is the probability of failure in one trial
n is the total number of trials
k is the number of successes we want to find the probability for
Now let's calculate the probabilities for each scenario:
a. She sells to all three.
In this case, we want to find the probability of getting 3 successes out of 3 trials (n = 3, k = 3). The probability of success in one trial is 0.2 (p = 0.2).
P(X = 3) = C(3, 3) * 0.2^3 * (1-0.2)^(3-3)
P(X = 3) = 1 * 0.008 * 1 (since (1-0.2)^0 = 1 when k = 3)
P(X = 3) = 0.008
So, the probability that she sells to all three is 0.008 or 0.8%.
b. She sells to exactly one.
In this case, we want to find the probability of getting 1 success out of 3 trials (n = 3, k = 1).
P(X = 1) = C(3, 1) * 0.2^1 * (1-0.2)^(3-1)
P(X = 1) = 3 * 0.2 * 0.64
P(X = 1) = 0.384
So, the probability that she sells to exactly one is 0.384 or 38.4%.
c. She sells to no one.
In this case, we want to find the probability of getting 0 successes out of 3 trials (n = 3, k = 0).
P(X = 0) = C(3, 0) * 0.2^0 * (1-0.2)^(3-0)
P(X = 0) = 1 * 1 * 0.512
P(X = 0) = 0.512
So, the probability that she sells to no one is 0.512 or 51.2%.