Probability and States

Question

1.     Suppose you have 2 duffel bags, each containing $1000 casino chips, some from Caesar's Palace and some from Harrah's (and nothing from anywhere else). In both bags 40% are real and the rest fake. In the first bag, 30% of the real chips are Caesar's and 10% of the fakes are Caesar's. In the second, 90% of the reals are Caesar's and 30% of the fakes are Caesar's. Would you rather have a random Caesar's chip from the first bag or the second? How about a random Harrah's chip from either bag? Explain your answers, including the probabilities you calculate in arriving at an answer. (I will assume that you prefer real chips to fake chips, no matter which casino they are from.) 

Answer 1

In bag 1:

40% of the chips are real, and of those, 30% are Ceasar's. 30% of 40% is 0.3*40% = 12%. So 12% of all the chips are real Ceasar's chips.

100%-40%=60% of the chips are fake, and of those, 10% are Ceasar's. 10% of 60% of 0.1*60% = 6%. So 6% of all the chips are fake Ceasar's chips.

Adding the real and fake Ceasar's chips, 18% of the chips in the first bag are Ceasar's. 12/18 = 2/3 of the Ceasar's chips are real.

In bag 2: 40% of the chips are real, and of those, 90% are Ceasar's. 90% of 40% is 0.9*40% = 36%. So 36% of all the chips are real Ceasar's chips.

100%-40%=60% of the chips are fake, and of those, 30% are Ceasar's. 30% of 60% of 0.3*60% = 18%. So 18% of all the chips are fake Ceasar's chips.

Adding the real and fake Ceasar's chips, 54% of the chips in the second bag are Ceasar's. 36/54 = 2/3 of the Ceasar's chips are real.

You can do the same thing, using the probabilities for Harrah's. For example, since 30% of the real chips in the first bag are Ceasar's, that means that 100%-30% = 70% of the real chips in the first bag are Harrah's. Then you follow the same procedure.

To finish answering the question, you will compare the final proportions for Ceasar's, and then compare them for Harrah's. You would rather have a random chip from the bag with the higher proportion of real chips.