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.
First, think carefully about the question. Do you assume the individual doesn't own one already? If two people meet, does it count if either buys, or only if the one you are following buys? What about if the individual you track has one, sells it, and buys another?
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 $(1-A)(1-(1-AB)^d)$
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