In a survey to determine the opinions of Americans on health insurers, 400 baby boomers and 600 pre-boomers were asked this question: Do you believe that insurers are very responsible for high health costs? Of the baby boomers, 232 answered in the affirmative, whereas 192 of the pre-boomers answered in the affirmative. If a respondent chosen at random from those surveyed answered the question in the affirmative, what is the probability that he or she is a baby boomer? A pre-boomer? (Round your answers to two decimal places.)
baby boomer ?
pre-boomer ?
Let events
B=baby boomer
~B=pre-boomer (complement of B)
A=answered in affirmative
then by the law of total probability:
P(A)=P(A∩B)+P(A∩~B)
=P(A|B)*P(B)+P(A|~B)*P(~B)
=232/400*(400/1000)+192/600*(600/1000)
=0.424
using Bayes theorem,
P(B|A)=P(A|B)P(B)/P(A)
=(232/400)*0.4/(0.424)
=0.5472
or
P(boomer|affirmative answer) = 0.5472
I'll leave it to you to find
P(pre-boomer|affirmative answer) by finding the complement.
Well, I'm no mathematician, but I can certainly try to come up with a response for you! Let's calculate the probabilities, shall we?
First, let's find the probability that a respondent who answered in the affirmative is a baby boomer. Out of the total 400 baby boomers surveyed, 232 answered in the affirmative. So, the probability of selecting a baby boomer is 232/400, which is approximately 0.58.
Now, let's calculate the probability that a respondent who answered in the affirmative is a pre-boomer. Out of the total 600 pre-boomers surveyed, 192 answered in the affirmative. So, the probability of selecting a pre-boomer is 192/600, which is approximately 0.32.
So, the probability that a respondent who answered the question in the affirmative is a baby boomer is 0.58, and the probability that they are a pre-boomer is 0.32.
Remember, these probabilities are just based on the given data. There could be other factors influencing the results, like someone pretending to be a boomer, but hey, who am I to judge? I'm just a clown bot, not an investigator!
To find the probability that a respondent chosen at random who answered in the affirmative is a baby boomer, we need to divide the number of baby boomers who answered in the affirmative by the total number of respondents who answered in the affirmative.
The number of baby boomers who answered in the affirmative is 232, and the total number of respondents who answered in the affirmative is 232 + 192 = 424.
So the probability that a respondent chosen at random who answered in the affirmative is a baby boomer is 232/424 = 0.55 (rounded to two decimal places).
To find the probability that a respondent chosen at random who answered in the affirmative is a pre-boomer, we need to divide the number of pre-boomers who answered in the affirmative by the total number of respondents who answered in the affirmative.
The number of pre-boomers who answered in the affirmative is 192, and the total number of respondents who answered in the affirmative is 232 + 192 = 424.
So the probability that a respondent chosen at random who answered in the affirmative is a pre-boomer is 192/424 = 0.45 (rounded to two decimal places).
Therefore, the probability that a respondent chosen at random who answered in the affirmative is a baby boomer is 0.55 and the probability that they are a pre-boomer is 0.45.
To find the probability that a respondent chosen at random is a baby boomer or a pre-boomer, we can use the concept of conditional probability.
First, we need to determine the total number of respondents who answered in the affirmative. This can be calculated by adding the number of baby boomers who answered yes (232) to the number of pre-boomers who answered yes (192).
Total respondents who answered yes = 232 + 192 = 424
Next, we can calculate the probability that a respondent chosen at random is a baby boomer. This can be found by dividing the number of baby boomers who answered yes (232), by the total number of respondents who answered yes (424).
Probability of being a baby boomer = number of baby boomers who answered yes / total respondents who answered yes
= 232 / 424
≈ 0.5472 (rounded to two decimal places)
Similarly, the probability of being a pre-boomer can be found by dividing the number of pre-boomers who answered yes (192), by the total number of respondents who answered yes (424).
Probability of being a pre-boomer = number of pre-boomers who answered yes / total respondents who answered yes
= 192 / 424
≈ 0.4528 (rounded to two decimal places)
Therefore, the probability that a respondent chosen at random is a baby boomer is approximately 0.55, and the probability that a respondent chosen at random is a pre-boomer is approximately 0.45.