# Probability when I have one or 3 choices

I'm wondering what is "better" to have in terms of profit:

Lets say we have 300 people come to your store and you have 3 products. Each individual is given (randomly) one product. If the individual likes the product he will buy it, but if he doesn't like the product, he will turn around and walk out of the store forever.

Now, I'm wondering is it better (more probable) to have 3 products or just one, in order to maximize the total number of sold products. Important to note is that customer doesn't decide which product he gets (he gets it randomly), he only decides if he likes it or not (50-50 chance he likes it).

Any help is welcome, or maybe link to some theory I should read in order to come p with a solution.

edit: Ok, so, a little detailed explanation: let say that I have unlimited number of each of the items in the store(currently 3 different items - but unlimited number of them in the stock), and lets say that each customer that comes in to my store (aproximatelly 1000 a day) either likes or doesn't like the randomly offered product to him. Each of the products offered is of the same popularity - we sold almost the same amount of product1, product2 and product3. Product1 sold for example just 100 more units than Product2 and Product 2 sold like 100 more units than Product3, but the sold numbers are as high as 1000000 so this difference is really very low. Does this now help in determining if we should chose only Product1 and keep forcing it, or should ve leave the three products as is?

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It rather depends on how popular (the probability of being liked) the products are. In your scenario it would be better to have the most popular of the three alone than all three, but better to have all three than just the least popular product. (You may want to adjust popularity to take into account any different profits per item sold, and look at expected profits per offered item.) But in your question they are all as popular as each other so it makes little difference whether you have one or three products. – Henry Nov 14 '11 at 13:07
If there's a straight 50% chance that a customer buys the product they're given, irrespective of the product, then surely it makes sense to give them all the most expensive product possible? – Chris Taylor Nov 14 '11 at 13:08

I think that you have a lot of unstated assumptions here. Especially the "walks out of the store forever" indicates that you are concerned about losing return customers without modelling this aspect.

Your situation seems to be more like a TV station that loses a viewer that is disgusted by your program.

I suspect that your actual model has to be something like this: If viewers see a program they hate, they will turn off your station forever, but they are willing to pay more for the subscription (or view more often, thus generating proportional advertisement revenue) if your station has a varied offer of programs they like.

The outcome will depend on the exact details. If you assume that people will view a time that is proportional to the acceptable programs offered, and viewers like your three choices independently with probability one half, then offering one program will make half of the population pay one unit, netting you $1/2$ unit.

Two programs will make a quarter of the population pay two units, netting you $1/2$ unit.

Three programs will make an eighth of the population pay three units, netting you $3/8$.

So, clearly, you should go with a single (kind of) program (which is probably cheaper than offering two).

And specialization is indeed seen in TV channels.

Note that the above is one model for your vague question, but the result is all about the details. You cannot apply mathematics before you have formulated a precise question and know what you want to model.

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+1 for the change of view. :) – J. M. Nov 24 '11 at 7:11

Well in my opinion this is a marketing matter. If it is possible, a market research about the most appealling products to the customers, would be the best proof of which product you should have in your store.

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is it better (more probable) to have 3 products or just one, in order to maximize the total number of sold products. Important to note is that customer doesn't decide which product he gets (he gets it randomly), he only decides if he likes it or not (50-50 chance he likes it).

Under the conditions you have described, the main objective is to maximize the total number of sold products. This number will be <= population number of 800.

The only factors here are: 1 - The number of people showing in the store 2 - The number of items you have in stock

factor 1 above is not described in your statement of the problem, but if you think that having more than 1 product will affect it then having 3 products is better than 1. Otherwise, the only factor would be factor (2) and adding new products will not add value.

I hope this helps.

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