# Getting Lost in (simple?) Bayes theorem application

Background: I have the following data:

27% of the respondents to a survey are interested in a product. Of that 27% which is interested:

• 59% are 15-24 years old.
• 18% are 25-34 years old.
• 6% are 35-44 years old.
• 4% are 45-54 years old.
• The rest are outside these age ranges.

The probability of being in each age range is:

• 11 % for 15-24.
• 13% for 25-34.
• 13% for 25-34.
• 12% for 34-45.
• rest is outside.

Problem: What percentage of each age group is interested in the new product?

Attempt: Bayes Theorem seems the right way to go here since I need to invert a probability: calling P(I) the probability of being interested in the product and P(E) the probability of being in a certain age range, from the survey I have P(I) (27% are interested) and I have P(E | I) and I want to find P(I | E).

So my attempt e.g. for the age group 15-24 was to do:

P(I|E ) = P(I) * P(E | I) / P(E) = 27% * 59% / 11%


27% is the overall probability of being interested in the product, 59% is the probability of being in age range 15-24 given being interested in the product.

Question: Calculating P(E) is where I get confused. I tried using 11%, which is how much of the population is in the age range 15-24.

But it does not work, because I get a number greater than 1 as a result.

What is the correct way to calculate P(E) in this problem and am I setting things up correctly ?

• Welcome to Cross Validated! Please take a moment to view our tour. Commented Mar 27, 2017 at 16:49

Vice versa, if the age ranges are those for respondents, then the numbers are clearly inconsistent. Proof: if $27\%$ of the respondents are interested in the product, and $59\%$ of those interested are aged 15-24, this means that there are at least $59\%\times.27\%\approx.16\%$ respondents aged 15-24 among all respondents. The reason is that these are only the respondents, aged 15-24, which are interested in the product: if there are other respondents, aged 15-24, which are not interested in the product, then the total percentage of respondents aged 15-24 must be even larger than $16\%$. However, the question text tells you that the percentage of respondents aged 15-24 is $11\%<16\%$, so something is clearly wrong here. Either the numbers are wrong, or you are not given the age groups for the respondents, but for the general population. However, in this case, the question should give you also the nonresponse bias by age, otherwise it's impossible to answer.
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