Independent Independent EventsEvents

These situations are dealing with These situations are dealing with Compound eventsCompound events involving two or involving two or more separate events.more separate events.

When will the occurrence of one When will the occurrence of one event effect the occurrence of event effect the occurrence of another?another?

Lets Play MONOPOLYLets Play MONOPOLY

GO TO JAIL!GO TO JAIL!• To get out of To get out of

Jail you must Jail you must roll doubles roll doubles on a pair of on a pair of standard standard dicedice

Lets look at the following QuestionsLets look at the following Questions

1. Determine the probability that you 1. Determine the probability that you will roll doubles on your first try.will roll doubles on your first try.

Let A represent the event that a double is Let A represent the event that a double is rolledrolled

You could get a double 1, 2,3,4,5 or 6 … This You could get a double 1, 2,3,4,5 or 6 … This gives you 6 possible outcomesgives you 6 possible outcomes

When you roll a dice there are 36 different When you roll a dice there are 36 different outcomesoutcomes

6 1( )

36 6P A or

2. On your first two turns you failed to 2. On your first two turns you failed to roll doubles. One of your roll doubles. One of your opponents’ reasons that on the next opponents’ reasons that on the next turn your odds of rolling doubles is turn your odds of rolling doubles is 50/50. Explain how your 50/50. Explain how your opponents’ reasoned this.opponents’ reasoned this.

-A be the even that you roll a double on your -A be the even that you roll a double on your first try. first try.

-B be the event that you roll a -B be the event that you roll a double on double on you second tryyou second try

-C be the event that you roll a double on your -C be the event that you roll a double on your third try. third try.

After two unsuccessful trials, the probability After two unsuccessful trials, the probability has risen to 3/6 or even odds.has risen to 3/6 or even odds.

1 1 1 3( )

6 6 6 6P AorBorC

3. What is the probability that you will 3. What is the probability that you will get out of jail on the third attempt?get out of jail on the third attempt?

( ) ( )

1( )

6

P rollsdoubles P A

P A

4. After how many turns is it certain 4. After how many turns is it certain that you will roll doubles? Explains.that you will roll doubles? Explains.

Infinite. Infinite.

There is no trial in which the There is no trial in which the probability is equal to one.probability is equal to one.

5. One of your opponents, explain that 5. One of your opponents, explain that each roll of the dice is an each roll of the dice is an independent event and that, since independent event and that, since the relatively low probability of the relatively low probability of rolling doubles never changes from rolling doubles never changes from trial to trial, you may never get out trial to trial, you may never get out of jail and may as well forfeit the of jail and may as well forfeit the game. Explain the flaw in this game. Explain the flaw in this analysis. What is correct in this analysis. What is correct in this analysis?analysis?

While the probability of rolling doubles on While the probability of rolling doubles on any one trial is relatively low, 1/6. There is any one trial is relatively low, 1/6. There is still a non-zero probability of the event still a non-zero probability of the event occurring. In fact, over a large number of occurring. In fact, over a large number of trials, we would expect to see doubles trials, we would expect to see doubles rolled once out of every six trials. rolled once out of every six trials.

Thus, you have a significant likelihood of Thus, you have a significant likelihood of getting out of jail before the game is over.getting out of jail before the game is over.

1 1 1 1 1 1 6( ) 1

6 6 6 6 6 6 6P AorBorCorDorEorF

Summary NoteSummary Note

In some situations involving compound In some situations involving compound events, the occurrence of one event events, the occurrence of one event has has no effect on the occurrence no effect on the occurrence of anotherof another

These events are These events are INDEPENDENTINDEPENDENT

Ex Ex

a)a) A coin flipped and turns up heads. A coin flipped and turns up heads. What is the Probability that the What is the Probability that the second flip will turn up heads?second flip will turn up heads?

a)a) A coin is flipped four times and A coin is flipped four times and turns up heads each time What is turns up heads each time What is the probability that the fifth will the probability that the fifth will turn up heads?turn up heads?

c) What is the probability of tossing c) What is the probability of tossing five heads in a row?five heads in a row?

The Compound Probability of The Compound Probability of two two independent independent events can be events can be

calculated by using the calculated by using the Product Product Rule for Independent EventsRule for Independent Events

( ) ( ) ( )P AandB P A P B

1 1 1 1 1( )

2 2 2 2 21

( )32

( ) 0.03125

P Hfivetimes

P Hfivetimes

P Hfivetimes

ExamplesExamples

1.1. A coin is flipped and a spinner with three A coin is flipped and a spinner with three equal sections (Red, blue and yellow) is equal sections (Red, blue and yellow) is spun. What is the probability of a head spun. What is the probability of a head and yellow?and yellow?

2.2. A calculator has a keyboard assembly and A calculator has a keyboard assembly and a logic circuit. The probability of a a logic circuit. The probability of a defective keyboard is 0.05 and the defective keyboard is 0.05 and the probability of a defective circuit is 0.02. probability of a defective circuit is 0.02. What is the probability that a calculator What is the probability that a calculator will not be defective when it is assembled?will not be defective when it is assembled?

Homework!Homework!

Pg 334Pg 334

#1,2,3,4ab#1,2,3,4ab