if 20% of the people in a community use the library in one year, find these probabilities.
a) at most 3 people used the library
b) exactly 3 people used the library
c) at least 5 people used the library
To find the probabilities for each scenario, we need to use the Binomial Probability Formula, which is:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
Where:
P(x) is the probability of achieving exactly x successes,
n is the total number of trials (people in the community),
x is the number of desired successes (people using the library),
p is the probability of success in one trial (percentage of people using the library),
(1-p) is the probability of failure in one trial.
Now let's calculate the probabilities for each scenario:
a) At most 3 people used the library:
We need to calculate the probabilities for 0, 1, 2, and 3 people using the library (since "at most" includes 0, 1, 2, and 3 people).
P(at most 3) = P(0) + P(1) + P(2) + P(3)
b) Exactly 3 people used the library:
P(exactly 3) = P(3)
c) At least 5 people used the library:
We need to calculate the probabilities for 5, 6, 7, ... up to n people using the library (since "at least" includes 5, 6, 7, ... up to n people).
P(at least 5) = P(5) + P(6) + P(7) + ... + P(n)
To calculate each probability, substitute the values into the Binomial Probability Formula for nCx, p, and (1-p), and compute the result for each scenario.