One of the books with a valued place on the shelves of the CarnotCycle library is *Introduction to Statistical Thermodynamics* by Malcolm Dole (1903-1990) who had a long and distinguished career at Northwestern Tech, and who incidentally was involved in the operations of the K-25 plant at Oak Ridge, TN. The Dole Effect, which led to carbon replacing oxygen as the reference standard for atomic weights, is named for him.

But back to his book. In the chapter on the math of probability, Professor Dole illustrates the multiplication and addition rules with this sampling problem:

*“A bag contains 2 white balls and 3 black balls, all balls being indistinguishable except for their colors. Three balls are drawn from the bag; what is the probability that the third ball is white?”*

OK, there are two ways to solve this problem. One way is to list the sequences in which three balls can be drawn ending with a white ball on the third draw. Letting W represent the drawing of a white ball and B represent the drawing of a black ball, we have

W – B – W

B – W – W

B – B – W

Next we apply the multiplication rule to calculate the probabilities of the three sequences

Finally we apply the addition rule to get the answer: 6/60 + 6/60 + 12/60 = 24/60 = 2/5. The probability that the third ball is white is 2/5. Problem solved. Now we can go look at Facebook.

Hey but hang on, you said there were two ways. What’s the other way?

The other way is difficult to put into words but can be described as the illusion of sampling, in the sense of appearing simply to take balls out of the bag. If you read the question again, you will notice that the colors of the first two balls drawn from the bag are not mentioned. Does this matter? Yes it does, because unless this information is supplied, the sampling procedure cannot be said to exclude putting the first and second balls back in the bag before drawing the third.

So the second way is to recognize this and calculate on the basis of sampling with replacement. Under these circumstances, at each draw there are always 2 white balls and 3 black balls in the bag and the probability of drawing a white ball is always 2/5. It’s a lot quicker to get the answer this way.

But wait a minute, the probability calculations above did not involve replacing any balls, so how come the answer turned out as 2/5?

The best way to see this is to visualize the entire set of sequences of drawing all five balls. Using the combinatorial math for permutation with repetition we see there are 5!/2!3! = 10 sequences in total. For simplicity, the diagram below shows only the white balls in each sequence.

1 |
2 |
3 |
4 |
5 |

O | O | |||

O | O | |||

O | O | |||

O | O | |||

O | O | |||

O | O | |||

O | O | |||

O | O | |||

O | O | |||

O | O |

end

What you are looking at here is the probability distribution of the two white balls in the absence of any information concerning the colors of balls drawn from the bag, which according to the principle established above is equivalent to sampling with replacement. Hence there must be four white balls in each column, representing the 4/10 = 2/5 probability of a white ball in that position. This is what the probability calculations given above are actually computing for row 2 (W-B-W), row 5 (B-W-W) and rows 8 and 9 (B-B-W) in column 3.

**– – – –**

**A playing card analogy**

Another way to illustrate the illusion of sampling is to replace the bag and the 2 white and 3 black balls with a pack of 5 playing cards of which 2 are red suits and 3 are black suits. Two cards are drawn from the pack and placed face down on the table. The question is now asked, what is the probability that the third card drawn is red? The answer is immediately 2/5 since the suits of the first two cards have not been revealed – under these circumstances all cards have the same 2/5 probability of being red. Dealing cards face down is therefore equivalent to sampling with replacement. It is not until the cards have been looked at that they can be considered by the person(s) doing the looking as a sample taken from the pack.

**– – – –**

P Mander April 2021