This post will go over probability theory. While not directly related to economics, probabilities and statistics are used in economics quite a bit. Here is the question giving people problems:
There are two bags, each containing 1000 ping-pong balls. Bag A contains
1000 red balls and no black balls, and bag B contains 200 red and 800
black. You are blindfolded and reach into a bag. There is a .5
probability it is bag A and a .5 probability it is bag B. You draw a red
ball. What is the probability that you drew the ball from bag A.
The math for doing probabilities goes like this. If you use the word AND you use X in the calculation.
If you use the word OR then you use + in the calculation.
For this particular problem you just need to multiply the probabilities by each other. Note the sentence:
You drew from bag A AND (X) a red ball.
So the probability of getting the first bag is 1/2, and the probability
of getting a red ball in the bag is 1000/1000 or 1, so multiplying them
together gives you 1/2 or 5/10. Let's go through the other scenarios:
You drew from bag A AND a black ball.
The probability of the first bag is 1/2, and the probability of getting a black ball from that bag is 0, so multiplying them together gives 0.
You drew from bag B AND a red ball.
The probability of the second bag is 1/2, and the probability of getting a red ball from that bag is 200/1000 or 1/5, and multiplying those together gives 1/10.
You drew from bag B AND a black ball.
The probability of the second bag is 1/2, and the
probability of getting a black ball from that bag is 800/1000 or 4/5, and
multiplying those together gives 4/10.
When we add all of the possibilities together we should get 1 (because we KNOW there will be an outcome) and we get 5/10 plus 0 plus 1/10 plus 4/10 added together gives us 1. So we know we did everything right.
So the final probability of this state will be 5/10 plus 4/10 or 9/10.
