Elementary Probability

Part 4: Independence

Suppose A and B are events in a sample space. The knowledge that an outcome is in event A may change your estimate of the likelihood that the outcome is in event B. For example, suppose that the experiment is rolling 2 fair dice, the event A is: The sum of the dice is greater than 10, and that the event B is: At least one of the dice is 5. Then knowing that the outcome is in the event A, increases the likelihood that the outcome is in the event B. Once we know that the outcome is in A, we have a new experiment with the sample space: {(5,6), (6,5), (6,6)}. Each of these outcomes is equally likely. So, the probability of one die being a 5 given that the sum is greater than 10 is 2/3.

On the other hand, if the experiment is flipping a fair coin twice, knowing that the first flip is a head (event A), does not change the likelihood that the second flip is a tail (event B). Here the new sample space is {(H,H),(H,T)}. Again, both outcomes are equally likely, so the probability that the second flip is a tail is 1/2 -- the same probability we would assign without the knowledge about A.

Informally, two events A and B are said to be independent if knowing that an outcome is in event A does not change the likelihood that the outcome is in event B and vice versa.

1. Suppose that you simultaneous flip a coin and roll a standard die. If you know that the die has come up 6, what is the probability that the coin shows heads? What does your answer tell you about the independence of the events of rolling a 6 and flipping a head?

2. Suppose that 50% of the people in a town are 5 ft. 7 in. or taller and that 50% of the people in this town are males. If a person from the town is chosen at random, are the events the person is taller than 5 ft. 7 in. and the person is a male likely to be independent? Explain.

The probability that a person chosen at random will be both taller than 5 ft. 7 in. and a male may at first appear to be 1/4 . After all, half the population is taller than 5 ft. 7 in. and half of that group is male, so 1/2 of 1/2 is 1/4. But since males on average are taller than females, it is wrong to assume that 1/2 of the 5 ft. 7 in and taller group is male. Knowing whether the person chosen is a male influences the likelihood (probability) that he is 5 ft. 7 in or taller.

Suppose that, for an experiment, the event A has probability 1/4 and event B has probability 1/3. If events A and B are independent, then in a large number of instances of this experiment, only 1/3 will be in the event A. Of this 1/3, only approximately 1/4 will also be in event B. So, the probability of the event A and B is just the product of the two probabilities, 1/12.

This observation is taken as the definition of the independence of two events.

Definition: Suppose we are considering the outcomes of a particular experiment. If A and B are events (subsets of the sample space of outcomes), we say A and B are independent if

3. In Question 2 of Part 3 you determined the 8 members of the sample space for flipping a fair coin 3 times. We list the elements of this sample space here:

   (H,H,H)	(H,H,T)	(H,T,H)	(H,T,T)
   (T,H,H)	(T,H,T)	(T,T,H)	(T,T,T)

   Let A be the event that a heads appears on the first flip, and let B be the event that a heads comes up on the third flip.

   1. Without calculating any probabilities, explain why the events A and B should be independent.

   2. Find P(A).

   3. Find P(B).

   4. Find P(A and B).

   5. Show that P(A and B) = P(A) P(B), and thus the events are indeed independent.

4. Again, consider the sample space obtained by flipping a fair coin 3 times. Let A again denote the event that the first flip is a head. Let C denote the event that at least 2 of the 3 flips are heads.
   1. Without calculating any probabilities, do you think that the events A and C are independent? Explain.

   2. Find P(A).

   3. Find P(C).

   4. Find P(A and C).

   5. Show that P(A and C) is not equal to P(A) P(C), and thus the events are not independent. Does this contradict your explanation in (a)?

5. A fair die is painted so that three sides are red, two sides are blue and one side is green. Thus, rolling the die has three possible outcomes R, B, and G.
   1. If the painted die is rolled once, what is the probability that it will come up blue?

   2. If the painted die is rolled twice, we can denote the nine possible outcomes by RR, RB, etc. Find the probability of each element in this sample space.

   3. Consider the following events in the sample space obtained by rolling the painted die twice.
      • A: At least one roll will be red.

      • B: The two rolls will have different colors.

      • C: Both rolls are red or both are blue.

      • D: Either both rolls are red or one roll is blue

      • E: At least one roll is red or at least one roll is blue.

      Among the events A, B, C, D, and E, determine which pairs are independent.

6. Suppose a fair die is rolled twice.
   1. Let A be the event that the first roll is greater than or equal to 2. Let A be the event that the second roll is greater than or equal to 4. Show that A and B are independent.

   2. Let A be the event that the first roll is greater than or equal to 2. Let C be the event that the sum of the rolls is greater than or equal to 4. Find P(A and C) and show that A and C are independent.

Independence and the Gambler's Fallacy.

A lack appreciation of the concept of independence lies at the heart of the mistaken belief that, after a run of bad luck, a gambler's luck is due to change. Click below for a short discussion of this misconception: