Dependent Dependent EventsEvents

Sometimes the outcome of an event Sometimes the outcome of an event can depend directly on the outcome can depend directly on the outcome of another event.of another event.

When this happens, the events are When this happens, the events are said to be said to be Dependent EventsDependent Events

Conditional ProbabilityConditional Probability

The Conditional Probability of an Event B. The Conditional Probability of an Event B.

Is the probability that B occurs GIVEN that A Is the probability that B occurs GIVEN that A has already occurredhas already occurred

( | )P B A

ExExA Professional hockey team has 8 Wingers. A Professional hockey team has 8 Wingers. Three of these wingers are considered very Three of these wingers are considered very good players or snipers. good players or snipers.

Every fall the team plays an exhibition match Every fall the team plays an exhibition match with the club’s farm team. In order to make the with the club’s farm team. In order to make the match more interesting for the fans, the match more interesting for the fans, the coaches agree to select two wingers at random coaches agree to select two wingers at random from the pro team to play for the farm team. from the pro team to play for the farm team.

What is the probability that the two pro wingers What is the probability that the two pro wingers that will play for the farm team will be the that will play for the farm team will be the snipers?snipers?

Let A represent the event that the first winger Let A represent the event that the first winger is a sniperis a sniper

Let B represent the event that the second Let B represent the event that the second winger is a sniperwinger is a sniperThree of the Eight are picked from the pro team. So the Three of the Eight are picked from the pro team. So the probability of a Pro winger being selected isprobability of a Pro winger being selected is

If the first winger selected is one of the good players, then If the first winger selected is one of the good players, then there are seven remaining wingers to choose from, two of there are seven remaining wingers to choose from, two of which are the good playerswhich are the good players

3( )

8P A

2( | )

7P B A

The Probability that 2 snipers will be selected The Probability that 2 snipers will be selected is is

3 2 6 3( ) 10.7%

8 7 56 28P AandB

Note: even though the events are Note: even though the events are dependent upon each other you dependent upon each other you can still multiply probabilities to can still multiply probabilities to

find the probability that they both find the probability that they both happen.happen.

Product Rule for Dependent EventsProduct Rule for Dependent Events

The Probability that dependent events will The Probability that dependent events will occur is given byoccur is given by

( ) ( ) ( | )

( )( | )

( )

P AandB P A P B A

P AandBP B A

P A

ExEx

Grace’s Computer sometimes crashes when Grace’s Computer sometimes crashes when she uses her Email Program. Her Email she uses her Email Program. Her Email program stops responding to command she program stops responding to command she can usually close the program without the can usually close the program without the computer crashing.computer crashing.

The Probability that her Email will not The Probability that her Email will not respond to commands is 2.5% and the respond to commands is 2.5% and the chance that both Email and computer crash chance that both Email and computer crash is 1%. If Grace’s Email is failing what is the is 1%. If Grace’s Email is failing what is the probability that the computer will crash?probability that the computer will crash?

Let A be the event that Graces Email failsLet A be the event that Graces Email fails

Let B be the event that the Computer Let B be the event that the Computer CrashesCrashes

Since event A triggers event B they are Since event A triggers event B they are dependent upon each other. The probability dependent upon each other. The probability of of the computer crashing depends on if the the computer crashing depends on if the Email Email failsfails

( )( | )

( )

0.01

0.0250.4

40%

P AandBP B A

P A

Summary NoteSummary NoteIf A and B are If A and B are independentindependent events then the events then the probability of both events occurring is given by probability of both events occurring is given by

If event B is If event B is dependentdependent upon event A, then the upon event A, then the conditional probability of B given A is P(B | A). conditional probability of B given A is P(B | A). In this case the probability of both events In this case the probability of both events

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

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

Homework!Homework!

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