Probability Question

Here's the question:

In a city there are 3 motorcycle sellers. Each seller make it's own motorcycle and sell it to people. In a day about 1000 motorcycles are sold in the city. The probability that a seller's bike would be sold is given by $P_n$ where n is the particular seller. Given $P_1 = 2/8$ , $P_2 = 4/8$ , $P_3 = 6/8$ , find the expected number of motorcycles sold by each seller.

I made this question. I am not sure if the question is technically correct or not. I further doubt if $P_1 + P_2 + P_3 = 1$ is a necessary condition or not.

Help me figuring it out.

Easy Math Editor

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Comments

It is consistent, but incomplete. $P_1 + P_2 + P_3 \neq 1$ is all fine; what we know is that "among the bikes that the first seller sells, it's expected that $P_1$ of them are actually sold", and so on. We're missing the number of bikes that each seller plans to sell. However, assuming the numbers are identical, we can solve it:

Suppose each seller makes $x$ motorcycles. Then seller $i$ is expected to sell $P_ix$ motorcycles. In total, there are $1000$ motorcycles sold, so $P_1x + P_2x + P_3x = \frac{3x}{2} = 1000$ , giving $x = \frac{2000}{3}$ . Thus the expected number of motorcycles sold by the first seller will be $P_1x = \frac{500}{3}$ , and similarly with other sellers.

Ivan Koswara - 7 years, 7 months ago

Oh thank you very much! It was easier then I expected; my teacher almost took the entire lecture and couldn't figure it out.

Lokesh Sharma - 7 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$