Lesson 3

Lesson 3: Independent and Dependent Events

Focus

There are many factors that can affect the outcome of a sporting event; because of this, specific rules are written for each sport. Home court advantage, travel distance, crowd size, and weather conditions are a few factors that may impact a team winning or losing.

Can you think of factors that have no bearing on the outcome of a game? In this lesson you will discover that there are some events that directly affect each other and others that are strictly independent.

Lesson Outcomes

At the end of this lesson, you will be able to

• determine if events are independent or dependent
• calculate the conditional probability of one event
• calculate the probability of two dependent or two independent events

Lesson Question

You will investigate the following question: How are problems that involve the probability of two independent or dependent events solved?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 3 Assignment (Download the Lesson 3 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms and Formula Sheet
• work under Project Connection

Discover

Finding the probability of two events can sometimes be challenging because the outcome of the second event may depend on the outcome of the first event.

In the following Try This you will investigate different probability scenarios.

Try This 1

Open Compound Independent and Dependent Events and complete the questions that follow.

1. Drag three blue and two green marbles into the bag. You will draw two marbles out by setting the “Number of Draws” to “2.” Replace the marbles after each draw by selecting “Yes.” Use a table similar to the following to record your responses.
   
   

   Experiment	Outcome	Calculation	Probability
   Two marbles are drawn and the first marble is replaced.	BB
   	BG
   	GB
   	GG

2. Now conduct the same experiment, but instead each marble is not replaced in the bag after it is drawn. Record your results in a table like the following.
   
   

   Experiment	Outcome	Calculation	Probability
   Two marbles are drawn and the first marble is not replaced.	BB
   	BG
   	GB
   	GG

3. What is the difference between the calculations of the probability when the marble is replaced versus when the marble is not being replaced?

Save your responses in your course folder.

Share 1

Work with a partner or in a group. Complete the following.

1. Discuss the pattern you found in the probability calculation when the marble is replaced and when the marble is not replaced.
2. Once you have identified a pattern, try another experiment with a different number of blue and green marbles. Does the same pattern hold true when you replace the first marble and when you do not replace the first marble?

If required, place a summary of your discussion in your course folder.

Explore

In Discover, you noticed the difference in probability when marbles were replaced versus when marbles were not replaced. Two events are independent if the outcome of one event has no influence on the outcome of the other event. With independent events, the number of possible outcomes remains the same throughout the experiment. Some examples of independent events include the following:

• You spin the spinner, like the one shown, twice. The result from the first spin is completely independent from the result of the second spin.
  
  
  
• Two cards are randomly selected from a standard deck of playing cards. After drawing the first card, it is returned to the deck. The second card is then drawn.
• Marbles are drawn from a bag and returned before the next draw.

Two events are dependent if the outcome of the first event influences the outcome of the second event. The number of possible outcomes is changed. Some examples of dependent events include the following:

• You call to make an appointment with the dentist and you select an appointment time. Your friend then calls the same dentist and selects an appointment time that is still available.
• At a fundraising event, a ticket is drawn. That ticket is awarded second prize and taped on the gift. Your ticket is still in the draw.
• Marbles are drawn from a bag and not returned before the next draw.

Watch the video “Identifying Independent Events” to see how independent or dependent events can be identified.

Self-Check 1

For independent events, you have seen that the probability of the second event is not influenced by the probability of the first. You will now investigate how to calculate the probability of two independent events.

Try This 2

Part A

You have a fair coin, a fair die, and a spinner.

1. Given the experiments in the table, calculate each probability. The first row is completed for you as a guide.
   
   

   Experiment	Event A	P(A) =	Event B	P(B) =	P(A) × P(B) =
   flipping a coin twice	flipping a head		flipping a tail
   flipping a coin and rolling a die	flipping a head		rolling a six
   spinning the spinner and flipping a coin	spinning a red		flipping a tail

Save your responses in your course folder.

Part B

Use Exploring Probability—Explore It to check your answers in Part A. Use either the tree diagram or the chart to check the probabilities of the independent events in your table from Part A.

2. After completing the table and checking your answers using the applet, describe in your own words how you would calculate the probabilities of two independent events.
   

Save your responses in your course folder.

From Try This 3, you can summarize calculating the probability of independent events as follows:

The formula states that the probability of two independent events can be determined by finding the product of the two events.

Watch the video clip titled “Independent Events 2” to further understand the use of the independent events formula.

Self-Check 2

The probability of independent events can also be calculated by using graphic organizers such as Venn or tree diagrams.

In Discover, you determined that in dependent events, the probability of the second event is influenced by the probability of the first event. This probability is considered to be conditional.

You can also have more than one event happening at the same time. Let’s investigate how to calculate the probability of two dependent events.

Try This 3

From Discover, you learned that when the marble was not replaced, the probability of the second event happening was directly affected and that the events were considered to be dependent. Use Compound Independent and Dependent Events to investigate how the probability of dependent events is calculated.

Start with a bag that contains three blue and four green marbles. Draw two marbles from the bag, one at a time. Each marble is not replaced in the bag after it is drawn. Then complete the activity using two blue marbles and seven green marbles in the bag. Do not replace marbles.

1. Using the applet, determine the calculation to find the probability of each outcome. Record your results in a chart like the following.
   
   

   Experiment	Outcome	Calculation
   three blue marbles and four green marbles	BB
   	BG
   	GB
   	GG
   two blue marbles and seven green marbles	BB
   	BG
   	GB
   	GG

2. How does the probability that the first event will occur directly affect the probability that the second event will occur? Explain.
3. Develop a general formula to find the probability for picking two blue marbles without replacing the first after it is drawn.
4. Explain why the probability of the second event is considered to be conditional.

Save your responses in your course folder.

Share 2

With a partner or in a group, compare the general formula that you created for question 3 of Try This 3. Come to an agreement on a general formula; then return to the applet and test the formula using a bag of two blue marbles and five green marbles.

If required, save a copy of your discussion in your course folder.

From Try This 4, you can summarize calculating the probability of two dependent events as follows:

P(B|A) is the notation for conditional probability. It should be read as “the probability of event B happening, given that event A has already occurred.”

The formula can be manipulated in the following ways:

Sometimes predictions are made that involve dependent events. In board games, for example, one player’s move depends on what an opponent has done. A player might have a next move all planned out, but an opponent can foil those plans with a preceding move!

Connect

Lesson 3 Assignment

Lesson 3 Summary

In this lesson you learned how to identify events that are independent and dependent. The act of replacing or not replacing during the first event plays an important role. Once the event is identified as dependent or independent, the appropriate formula is used to solve probability problems.

You can apply independent and dependent formulas to scenarios that can affect the outcome of a game; for example, home court advantage, travel distance, crowd size, weather conditions, and other factors that may impact a team winning or losing.