the television show Law & Disorder has been successful for many years. That show recently had a share of 20, which means, that among the TV sets in use, 20% were tuned to Law & Disorder. An advertiser wants to verify that 20% share value by conducting its own survey, and a pilot survey begins with 11 households have TV sets in use at the time of a Law & Disorder broadcast.
Find the probability that none of the households are tuned to Law & Disorder.
P(none) =
Find the probability that at least one household is tuned to Law & Disorder.
P(at least one) =
Find the probability that at most one household is tuned to Law & Disorder.
P(at most one) =
If at most one household is tuned to Law & Disorder, does it appear that the 20% share value is wrong? (Hint: Is the occurrence of at most one household tuned to Law & Disorder unusual?)
yes, it is wrong
no, it is not wrong
To find the probabilities in this scenario, we need to understand a few concepts and use basic probability calculations.
1. Probability that none of the households are tuned to Law & Disorder (P(none)):
This can be calculated using the complement rule, which states that the probability of an event occurring is equal to one minus the probability of the event not occurring. In this case, the event would be that none of the households are tuned to Law & Disorder.
Since we have 11 households, and each household has a 20% chance of being tuned to Law & Disorder, the probability of a household not being tuned to Law & Disorder is 1 - 0.20 = 0.80.
To find the probability for all 11 households not being tuned to Law & Disorder, we multiply this probability for each household: P(none) = 0.80^11 ≈ 0.0693, or approximately 6.93%.
2. Probability that at least one household is tuned to Law & Disorder (P(at least one)):
To find this probability, we can use the complement rule again. The event here is that at least one household is tuned to Law & Disorder.
Using the same reasoning as before, the probability of a household not being tuned to Law & Disorder is 0.80. Therefore, the probability of having no households tuned to Law & Disorder is 0.80^11 (as we calculated in the previous step). Subtracting this value from 1 will give us the probability of having at least one household tuned to Law & Disorder: P(at least one) = 1 - 0.80^11 ≈ 0.9307, or approximately 93.07%.
3. Probability that at most one household is tuned to Law & Disorder (P(at most one)):
To find this probability, we simply need to calculate the probability of having one household tuned to Law & Disorder, and the probability of having none. We can then add these two probabilities together.
From step 1, we know that the probability of none of the households being tuned to Law & Disorder is 0.0693. To find the probability of exactly one household being tuned to Law & Disorder, we can multiply the probability of one household being tuned (0.20) by the probability of the others not being tuned (0.80^10, as there are now 10 households left). This gives us P(one) = 0.20 * 0.80^10 ≈ 0.2684.
Therefore, P(at most one) = P(none) + P(one) = 0.0693 + 0.2684 ≈ 0.3377, or approximately 33.77%.
Finally, to determine whether the 20% share value is wrong, we can compare the calculated probabilities with the expected value of 20%. Since the probability of at most one household being tuned to Law & Disorder is relatively low (approximately 33.77%), it suggests that the actual share value may not be accurately reflected by the survey results. Thus, it implies that the 20% share value might be wrong.