Good businessman turns bad - a probability problem

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Avik Roy
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Good businessman turns bad - a probability problem

Unread post by Avik Roy » Thu Jan 27, 2011 10:55 pm

Consider that a businessman has gained quite a good will and managed $X$ regular customers who will buy precisely one of his product each month. Now the businessman turns greedy and decides to make some quick profit. He brings $50\%$ bad products and mixes it randomly with the good products so that the product sold to any customer may be equally good or bad. A bad product might prove satisfactory with a probability of $p_1$. It is known that a client will stop buying products from that businessman if
-he finds one product unsatisfactory with probability $p_2$,
-he finds two products unsatisfactory with probability $p_3$ and
-he receives three products with a probability $1$
What is the expected number of customers left after three months?
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

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Re: Good businessman turns bad - a probability problem

Unread post by abir91 » Fri Jan 28, 2011 3:03 pm

Maybe you mean three unsatisfactory product in the third line?
Let, $P(i)$ = Probability of getting exactly $i$ bad products. Then,

\[ P(0) = \frac{1}{8}, \; P(1) = \frac{3}{8}, \; P(2) = \frac{3}{8}, \; P(3) = \frac{1}{8} \]

So, probability of finding $k$ products unsatisfactory, $S_k = \displaystyle\sum_{i=1}^{i=3} \binom{i}{k}P(i){p_1}^{i-k}(1-{p_1})^{k}$

Hence, the probability that a customer will leave is, $T = p_2 S_1 + p_3 S_ 2 + S_3$. So, the expectation is TX.

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