The probability that a person has an unlisted telephone number is 0.15. The district manager of
a political action group is phoning people urging them to vote. In a district with 416 households, all households have phones. What is the probability that the district manger will find that:
(a) 50 or fewer households in the district will have unlisted numbers?
(a) 345 or more have listed phone numbers?
(c) between 40 and 80 have unlisted numbers?
To find the probabilities in this scenario, we need to use the binomial probability formula. The binomial probability formula is:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Where:
- P(x) is the probability of having exactly x successes.
- n is the number of trials or households in this case.
- p is the probability of success (having an unlisted phone number).
- C(n, x) is the number of combinations of n items taken x at a time.
Now let's solve each part of the question:
(a) To find the probability that 50 or fewer households in the district will have unlisted numbers, we need to find the cumulative probability from x=0 to x=50.
P(50 or fewer households) = P(x=0) + P(x=1) + ... + P(x=50)
To find each individual probability, we'll plug in the values into the binomial probability formula.
P(x=0) = C(416, 0) * 0.15^0 * (1-0.15)^(416-0)
To find the combinations C(416, 0), we can use the formula: C(n, x) = n! / (x! * (n-x)!), where "!" represents factorial.
P(x=1) = C(416, 1) * 0.15^1 * (1-0.15)^(416-1)
Repeat this process for each value of x up to 50, and then sum them all together to get the probability.
(b) To find the probability that 345 or more households have listed phone numbers, we need to find the cumulative probability from x=345 to x=416.
P(345 or more households) = P(x=345) + P(x=346) + ... + P(x=416)
Follow the same process as in part (a) to find each individual probability and sum them up.
(c) To find the probability that between 40 and 80 households have unlisted numbers, we need to find the cumulative probability from x=40 to x=80.
P(40 to 80 households) = P(x=40) + P(x=41) + ... + P(x=80)
Again, follow the same process to find each individual probability and sum them up.
Note: Calculating binomial probabilities can be time-consuming when there are many trials involved. In such cases, using statistical software or a calculator with a binomial probability function can make the calculations easier.