7.5 Conditional Probability and Independence



This tutorial: Part C: Independent Events

Here is a little warm-up quiz, based on the formula for conditional probability in Part A of this tutorial:

Which of the following are true for arbitrary events A, B, and C? (Note: More than one may be true.)

Independent Events

The events A and B are independent if any one of the following three equivalent conditions hold:

P(A \cap B) = P(A)P(B)
P(A\|B) = P(A) B has no effect on A
P(B\|A) = P(B) A has no effect on B

Intuitively, two events are independent if the occurrence of one has no effect at all on the probability of the other.

If two events A and B are not independent, then they are dependent.

Examples

1. If there is a 10% chance that Jupiter will align with Mars, and a 50% chance that your coin flip will be heads, then what is the probability that Jupiter will align with Mars and your coin flip will be heads (assuming that Jupiter has no influence on your coin flip)?

Here,

    J: Jupiter aligns with Mars
    H: Your coin flip is heads
Since Jupiter has no effect on your coin flip, we take these events as independent, and so the probability that both events will occur is
    P(J \cap H) = P(J)P(H) = (.10)(.50) = .05.

2.
               

3. You roll a pair of dice two times in a row, and each time you add the numbers facing up. What is the probability that you roll a twice in succession?
             
Answer must be accurate to at least 4 decimal places, or it can be entered as a formula instead.

Test for Independence

To test whether two events A and B are independent, calculate P(A), P(B), and P(A \cap B), and then check whether P(A \cap B) equals P(A)P(B). If they are equal, A and B are independent; if not, they are dependent.

Examples

1. You throw two fair dice, one green and one red, and observe the numbers uppermost. Take A: the sum is 7, and B: the red die comes up even. Are these two events independent?
Solution Take

    A: The sum is 7; P(A) = \frac{n(A)}{n(S)} = \frac{6}{36} = \frac{1}{6}
    B: The red die shows an even number; P(B) = \frac{n(B)}{n(S)} = \frac{18}{36} = \frac{1}{2}
    A \cap B: The sum is 7 and the red die is even; P(A \cap B) = P\{\.(1, 6), (3, 4), (5, 2)\.\} = \frac{3}{36} = \frac{1}{12}

Test for Independence:

    P(A \cap B)=P(A)P(B)?
    \frac{1}{12}=\frac{1}{6} \frac{1}{2}

Therefore, the events are independent.

The following quiz is similar to Exercises 31-36 in Section 7.5 of or .

You throw two fair dice, one green and one red, and observe the numbers uppermost. Decide which of the following pairs of events are independent.

You now have several options:

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Last Updated: July, 2009
Copyright © 2009 Stefan Waner