New Information, Reduced Sample Space, and Conditional Probability

To illustrate the conditional nature of probabilities, suppose you have a box that has 6 slips of paper – the slips are labeled with the numbers 2, 4, 6, 8, 10, and 12. You select two slips at random from the box.

We assume that we’re sampling without replacement and the order that we select the slips is not important. Then we can list all of the possible outcomes. (By the way, since we are choosing two numbers from six, the total number of outcomes will be .)

S = {(2, 4), (2, 6), (2, 8), (2, 10), (2, 12), (4, 6), (4, 8), (4, 10), (4, 12)

(6, 8), (6, 10), (6, 12), (8, 10), (8, 12), (10, 12)}.

Suppose we are interested in the probability the sum of the numbers on the two slips is 14 or higher. Assuming that the 15 outcomes we listed above are equally likely, we see there are 9 outcomes where the sum is 14 or higher and so

P(sum 14 or higher) = 9/15.

Next, suppose we’re given some new information about this experiment -- both of the numbers on the slips are single digits. Given this information, we now have only six possible outcomes. We call this new sample space the reduced sample space based on the new information.

S = {(2, 4), (2, 6), (2, 8), (4, 6), (4, 8), (6, 8)}

We can evaluate the probability Prob(sum is 14 or higher) given that both of the slip numbers are single digits. Since there is only one way of obtaining a sum of 14 or higher in our new sample space, we see

P(sum of 14 and higher) = 1/6.

NOTATION: Suppose that E is our event of interest and H is our new information. Then we write the probability of E given the new information H as Prob (E | H), where the vertical line “|” means “conditional on” or “given” the new information. Here we found P(sum is 14 or higher | both slip numbers are single digits).

How does the probability of “14 or higher” change given the new information? Initially, the probability of 14 and higher was pretty high (9/15), but given the new information, the probability dropped to 1/6. Does this make sense? Yes. If we are told that both numbers are single digits, then we have drawn small numbers and that would tend to make the sum of the digits small.

Practice: new information, reduced sample space, and conditional probability

Let’s consider the familiar example where we roll a red die and a white die – the 36 possible outcomes of rolling the dice are shown in the table below.

Consider the following three events:

S = the sum of the two rolls is 7

E = the red die is an even number

D = the rolls of the two dice are different

		Roll on red die
		One	Two	Three	Four	Five	Six
Roll on white die	One	X	X	X	X	X	X
	Two	X	X	X	X	X	X
	Three	X	X	X	X	X	X
	Four	X	X	X	X	X	X
	Five	X	X	X	X	X	X
	Six	X	X	X	X	X	X

1. Find the probability you roll a sum equal to 7, that is, P(S).

2. Suppose we are told that the red die is an even number (event E). Circle the outcomes that correspond to the reduced sample space given this information. How many outcomes are there in this reduced sample space?

3. Find the probability of that the sum is equal to 7 given this information, that is, P(S | E).

4. Compare your answers to 1. and 3. Does knowing the red die is even change your probability of rolling a seven?

5. When the knowledge of one event does not change the probability of a second event, then we say that the two events are independent. Are events S and E independent?

6. In a similar fashion, compute P(S) and P(S | D) and compare the two probabilities. Are events S and D independent?