The television show Ghost Whistler has been successful for many years. That show recently had a share of 23, which means, that among the TV sets in use, 23% were tuned to Ghost Whistler. An advertiser wants to verify that 23% share value by conducting its own survey, and a pilot survey begins with 13 households have TV sets in use at the time of a Ghost Whistler broadcast.
Find the probability that none of the households are tuned to Ghost Whistler.
P(none) =
Find the probability that at least one household is tuned to Ghost Whistler.
P(at least one) =
Find the probability that at most one household is tuned to Ghost Whistler.
P(at most one) =
If at most one household is tuned to Ghost Whistler, does it appear that the 23% share value is wrong? (Hint: Is the occurrence of at most one household tuned to Ghost Whistler unusual?)
yes, it is wrong
no, it is not wrong
this is binary, like the other problem ... t = tuned in ... n = not tuned in
... p(t) = .23 ... p(n) = .77
(n + t)^13 = n^13 + 13 n^12 t + ... + 13 n t^12 + t^13
P(none) = n^13 = .77^13
P(at least one) means not none ... P(at least one) = 1 - P(none)
P(at most one) means one or none ... the sum of the 1st two terms
... (.77^13) + (13 * .77^12 * .23)
To find the probability that none of the households are tuned to Ghost Whistler, we can use the concept of independent events. Since each household's decision to tune in or not is independent of each other, the probability that each household is not tuned to Ghost Whistler is 1 - 0.23 (complement of 23%). Therefore:
P(none) = (1 - 0.23)^13
To find the probability that at least one household is tuned to Ghost Whistler, we can again utilize the concept of independent events. The probability of at least one household being tuned to Ghost Whistler is the complement of none of the households being tuned to Ghost Whistler. Therefore:
P(at least one) = 1 - P(none)
To find the probability that at most one household is tuned to Ghost Whistler, we need to calculate the probabilities of none and only one household being tuned to Ghost Whistler and then sum them up. Therefore:
P(at most one) = P(none) + P(only one)
Now, let's analyze if the share value of 23% is wrong or not. If the occurrence of at most one household being tuned to Ghost Whistler is unusual, then it suggests that the actual share value of 23% is incorrect. However, if the probability of at most one household being tuned to Ghost Whistler is reasonably high, it suggests that the 23% share value is accurate.
To make a conclusion, we need to compare the calculated probability of at most one household being tuned to Ghost Whistler with a reasonable threshold. If the probability exceeds this threshold, we can conclude that the 23% share value is accurate.
To find the probabilities, we need to understand the total number of possible outcomes and the number of favorable outcomes.
For this problem, we are given that the pilot survey consists of 13 households. We want to find the probability that none of the households are tuned to Ghost Whistler.
To find the number of favorable outcomes, we need to calculate the number of ways none of the households can be tuned to Ghost Whistler. Since each household can either be tuned or not tuned, and we want none of them to be tuned, we have only one favorable outcome.
The total number of possible outcomes is given by the total number of ways each household can be tuned or not tuned, which is 2 raised to the power of the number of households in the survey (2^13).
Therefore, P(none) = number of favorable outcomes / total number of possible outcomes = 1 / (2^13) = 1 / 8192.
To find the probability that at least one household is tuned to Ghost Whistler, we can subtract the probability that none of the households are tuned from 1. So, P(at least one) = 1 - P(none) = 1 - 1 / 8192 = 8191 / 8192.
To find the probability that at most one household is tuned to Ghost Whistler, we need to calculate the probability that none or exactly one household is tuned. We already have the probability of none (P(none)), and to find the probability of exactly one, we can multiply the probability of one household being tuned (1 / 8192) by the number of households (13). Then we add the two probabilities together.
P(at most one) = P(none) + (13 * (1 / 8192)) = 1 / 8192 + 13 / 8192 = 14 / 8192 = 7 / 4096.
Now, let's determine if the 23% share value of Ghost Whistler is wrong by considering the probability of at most one household being tuned. If this probability is significantly different from 23%, then it suggests that the share value might be wrong.
Given that the probability of at most one household being tuned is 7 / 4096, which is significantly smaller than 23%, it indicates that the 23% share value might be wrong. Therefore, the answer is yes, it is wrong.
Note: The occurrence of at most one household tuned to Ghost Whistler is unusual compared to the expected share value of 23%.