Learning Resources

Lesson

A bag has 50 colored marbles in it, of which 5 are red.  Event A is "drawing a red marble from the bag". We know from our previous work that the probability of A is calculated as follows:
                                        

Suppose event B is "tossing a head on a fair coin".  We know from our previous work that the probability of A is calculated as follows:
                                                

In these examples, event A in no way affects event B. Whether you draw a red marble or not, the probability of tossing a head is still ½. Events A and B are said to be independent events since event A does not change the probability of event B.

However, suppose event B is "drawing a second red marble from the same bag". Now the probability of this event depends on whether the first marble was placed back in the bag or not. If it were not replaced, then the "total number of possible outcomes" has been reduced to 49 and this will change the probability of event B. Events A and B are said to be dependent events since event A changes the probability of event B. This generally happens when there is no replacement of the item to the sample set.

Example

You are dealt a hand of 5 cards from a regular deck of 52 playing cards. Four of them are hearts and the other a spade. You discard the spade and ask for another card. What is the probability the new card you receive is a heart? Is this an example of dependent or independent events?

Solution

Since you have 5 cards, there are only 47 remaining from which to choose. The "total number of possible outcomes" (cards left) is 47. Also, since you have 4 hearts there are only 9 left in the deck. The "number of ways of the desired event occurring" (red cards left) is 9. The probability the new card is a heart is:
                                            

Dependent and independent does not apply since there is only one event involved in this example, viz. "receiving a heart". You just have to be careful how you count the total and desired outcomes.

Activity

1. Complete Investigation 4 on pages 310 & 311 in your text.
2. Complete the Investigation Questions 13 - 15 on page 311.
3. Do the CYU Questions 16 - 20 on pages 311 & 312.

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

Test Yourself

1. One fair coin is flipped 2 times. Event A is "a head occurs on the first flip" and event B is "a head occurs on the second flip". Are A and B dependent or independent? Explain.
2. An urn contains 4 balls numbered 1, 2, 3, 4 respectively. Two balls are drawn without replacement. Event C is "the first ball drawn has a 1 on it" and event D is "the second ball drawn has a 1 on it". Are C and D dependent or independent? Explain.

Solution