Probability question based on 2 distributors of an object

Question

Problem statement:

A shop is provided by $2$ distributors of keyboards. From the first distributor the shop buys $21$ normal keyboards and $14$ wireless keyboards, and from the second one $11$ normal keyboard and $22$ wireless keybaords. Knowing that the shop sold a wireless keyboard, which is the probability that, that keyboard was from the second distributor?

Let

$A_1$ = "keyboard sold from the first distributor"

$A_2$ = "keyboard sold from the second distributor"

$B$ = "keyboard is wireless"

Then we need to find $P(A_2 | B)$

In my textbook it says that $P(A_1) = P(A_2) = \frac 12$ But i think it doesn't make sense since there are more keyboards from the first than from the second.

Answer 1

Your textbook is absolutely correct. What that statement means is that, before having any knowledge of what kind of keyboard each shop sells and how many of each and which one you received. The probability that the keyboard came from any of the shops is equally likely.

Once you receive your keyboard and enquire about what kind of keyboard each shop sells and how many it has of each type. You can update your initial probabilities of a 50-50 chance using this information. Which is what you’re asked to do.

Answer 2

You're right in your confusion. It is am ambiguous statement to make. The event $A_1$ and $A_2$ are not very clearly defined. The two interpretations for $A_1$ (and similarly $A_2$) are:

1. You have keyboards in your store, some of which are from Seller 1 and some of which are from Seller 2. What is the Probability that when you pick a keyboard, it is from Seller 1?

In this case $P(A_1) = \frac{35}{68}$.

2. What is the probability that the seller $1$ sold you a keyboard?

In this case it may be considered, for the sake of simplicity by removing factors such as who is a better salesman or which is offering you a better price/product range, that $P(A_1) = 0.5$.

So I'd suggest not worrying about it too much because the statements truly are a little ambiguous.