Probability of shoppers buying specific items

First, let me thank any in advance who are able to offer assistance. I am having some challenges with a two part question determining the probability of shoppers buying specific items. Below are the questions and what I have worked out so far:

7598 people were asked about their recent shopping habits. 1496 people said that they had bought a designer purse in the last year. 551 people said that they had bought designer sunglasses in the last year. 6282 said that they had bought neither a designer purse nor designer sunglasses.

- How many people bought at least one of the two?

The total surveyed- total with no purchase= total purchasers of at least one

7598-6282= 1316

-How many people bought a designer purse AND designer sunglasses?

The second part is what confuses me. This is where I am having problems. I cannot seem to isolate either the sunglasses or the purses to get grasp on where the overlap is. I know that this cannot exceed 1316 people total, but I don't know where to start.

Re: Probability of shoppers buying specific items

Are you sure those numbers are correct, as you seem to have a contradiction... You note that 6282 people bought nothing, while 1496 people bought a purse, even if we assume that no one bought only sunglasses (i.e. if you bought sun glasses, you also bought a purse), then the sum of people who bought something and people who didn't is larger than your stated sample size. Note also, that you calculate (logically) "total purchasers of at least one" to be 1316, but you already know that 1496 people bought at least a purse.

Re: Probability of shoppers buying specific items

Nice catch. Maybe some people who bought a purse later returned it, so they are counted in two categories: as those who bought and those who didn't. (Smile)

Re: Probability of shoppers buying specific items

After re-reading the question and sharing the concern that the numbers do not add up (where my initial confusion stemmed from), I found that the printed material was transposed and that the correct total of survey members is 7958.

Therefore, can I correctly solve this as the following? I believe that the first part would be 7958-6282= 1676 bought at least 1 of the items.

In order to find out how many bought both, I would add the number of sunglasses to the number of purses (551+1496) and then subtract those that bought at least one (1676)?

So, finding 551 bought sunglasses, 1496 bought purses, and 371 bought both?

Re: Probability of shoppers buying specific items

Quote:

Originally Posted by

**tizpan** In order to find out how many bought both, I would add the number of sunglasses to the number of purses (551+1496) and then subtract those that bought at least one (1676)?

So, finding 551 bought sunglasses, 1496 bought purses, and 371 bought both?

Yes, that's correct. In particular, if you were to draw a Venn diagram of the scenario where set $\displaystyle A$ is the number of purses, and set $\displaystyle B$ the number of glasses, and label the intersecting region $\displaystyle x$, then you can determine the value of $\displaystyle x$ by noting that the number of people who bought only purses is $\displaystyle (A-x)$, the number of people who bought only sunglasses is $\displaystyle (B-x)$, and the total number of people that bought at least one item is (as calculated) 1676. This then leads to

$\displaystyle (A-x)+(B-x)+x=A+B-x=1676$

which is ultimately what you did.