# Companies and product probability question

Given set of people and company $X$.Company $X$ is new and promoting its new product in market.They are giving few product free to some people from set. Let the probability of an individual getting product at the start be $A$. Let the probability of buying a product in any given meeting with who already have product is $B$. What is probability of an individual buying product in $d$ random meetings with individuals from population.

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Let's assume that the individual we are tracking might own one, won't buy a second, and we don't count if he sells it. Start with just one meeting. You need the individual not to have the product, which has probability $1-A$, the one he meets to have one, probability $A$, and to have a purchase decision, probability $B$. The chance of a purchase is the probability of these $(1-A)AB$
For more meetings, assuming our target starts out without one (we can multiply by $1-A$ at the end) it is easier to calculate the chance of never buying and subtract from $1$. We had the chance of buying at $AB$, so the chance of not buying is $1-AB$. The chance of not buying in $d$ tries is then $(1-AB)^d$, so the chance of buying is $1-(1-AB)^d$. We said he wouldn't buy a second, so the overall chance of buying in $d$ tries is $(1-A)(1-(1-AB)^d)$
@user997704: First comment-that is exactly what happens in my second paragraph. To have a sale, you need one person to have one, one not to, and $B$ chance of a sale then. Second comment-I took it as $A$. A fraction $A$ of the population started with them and sales just change who has them, not how many there are. So a random person has chance $A$ to have one. – Ross Millikan Apr 14 '12 at 15:41