Probabilities and drawing ping pong balls - FreeEconHelp.com, Learning Economics... Solved!

3/9/12

Probabilities and drawing ping pong balls

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.

If we wanted to know the probability of drawing from bag A or drawing a black ball we would have to add the probabilities together.  The probability of drawing from bag A is 1/2, and the probability of drawing a black ball is 4/10 (from adding 0 to 4/10 from above).  

So the final probability of this state will be 5/10 plus 4/10 or 9/10.

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