# non-normalized probabilities question

In my intro probability/stats class, we were given the following problem: A restaurant owner would like to know how frequently his guests attend his restaurant. Based on an online survey, he is given the following information:

• 80% went in the last 3 months
• 48% went in the last month
• 25% went in the last 2 weeks
• 14% went in the last 7 days

The problem is: say 1000 people attend his restaurant. How many of these 1000 attended in the last 7 days, 2 weeks, month, and 3 months.

I thought initially that this was a simple multiply 1000 by the percentages above, but as you can see, they do not add to 100. My secondary thought was to normalize these perhaps, so it's not 80%, but 0.80 / (0.80 + 0.48 + 0.25 + 0.14), not 48%, but 0.48/(0.80 + 0.48 + 0.25 + 0.14), and so forth, so you get: [0.47, 0.287, 0.1497, 0.08], which now do add to 100, and you get some sensible answers. But, I don't know why this should work, or if it is the correct approach?

• Why do you suppose the answers ought to sum to 1000? What would such a summation mean? "Five people visited me this week, one on Monday, one on Tuesday, one on Wednesday, one on Thursday, another on Friday. Thus, 80% visited me Tuesday through Friday, 60% on Wednesday through Friday, and 20% on Friday. Wow, I had a lot of visitors: 80%+60%+20%=160% of them!" – whuber Sep 28 '17 at 18:22

## 1 Answer

These events are not exclusive: all the ones who came in the last two weeks also came in the last month. That why the probabilities increase as the period increases. They don't add to 100%. Instead, they tend to 100% as the period increases: 100% of previous customers visited the restaurant since its opening date.

• Welcome to our site, Hugh. Your post shows a good way to answer homework problems: by providing insight and explanation, while leaving responsibility for developing the answer to the original poster (OP). (For more about this, if you're curious, see the information at our self-study tag wiki.) – whuber Sep 28 '17 at 19:34