conditional probability 3.2 raise your hand when you are finished reading

CONDITIONAL PROBABILITY3.2

Raise your hand when you are finished reading.

The probability of an event occurring, given that another event has already occurred. The conditional probability of event B occurring, given that event A has occurred, is denoted by P(B/A)

Conditional Probability

STARTING SIMPLE

There are more red coins among the large coinsthan among the small coins. When randomly selecting a coin from the box, the probability of obtaining a red coin is what?

P(red) = 5/10 = 1/2 because there are 5 red coins in a total of 10 coins.

NOW ASSUME THAT A COIN IS PICKED BY ABLINDFOLDED PERSON.

The coin can be felt to be large, but the color is unknown.

What is the probability that the coin is red and large?

IN THIS CASE P(RED AND LARGE) = 3/5

Because all the small coins are ruled out and only the large coins are considered. There are 5 large coins, 3 of which happen to be red.

DEPENDENT

The probability of red changed when it was knownthat the coin was large.

P(red and large) P(red)

This is an indication that color and size are NOT independent.

They are DEPENDENTIf P(B/A)P(B) or when P(A/B)P(A)

INDEPENDENT

Two events are independent when the occurrence of one of the events does not affect the probability of the occurrence of

the other event. Two events A and B are independent when

P(B/A)=P(B) or when P(A/B)=P(A)

INDEPENDENT

P(B/A)=P(B) or when P(A/B)=P(A)“the probability of B, given A”

A is the event that happened firstB is the event that is happening second

Because they are independent The first event does not affect the Probability of the second event.

ROLL THE DICE

Roll the first Dice then roll the second dice

1.) What is the probability that you will roll the same number again?

2.) What is the probability that you will roll an even number?

DEPENDENT OR INDEPENDENT

1.) Returning a movie after the due date and receiving a late fee.

2.) A father having hazel eyes and a daughter having hazel eyes.

3.) Select a 10 from a deck of cards, replacing it, and then selecting a 10 from the deck again.

THE MULTIPLICATION RULE

The probability that two events A and B will occur in sequence is

P(A and B) = P(A)P(B/A)

If events A and B are independent, then the rule can be simplified to

P(A and B) = P(A)P(B)

USING THE MULTIPLICATION RULE

Two cards are selected, without replacing the first card, from a standard deck. Find the probability of selecting a king and then selecting a queen.

P(K and Q) = P(K)P(Q/K)= =

USING MULTIPLICATION RULE

To Find Probabilities

For anterior cruciate ligament (ACL) reconstructive surgery, the probability that the surgery is successful is 0.95.

Find the probability that three ACL surgeries are successful.

EXAMPLE 1

The probability that each ACL surgery is successful is 0.95. The chance of success for one surgery is independent of the chances for the other surgeries.

P(three surgeries are successful)=(0.95)(0.95)(0.95)

The probability that all three surgeries are successful is about 0.857.

EXAMPLE 2

To Find Probabilities


2.) Find the probability that none of the three ACL surgeries are successful.

EXAMPLE 2 CONTINUED

Because the probability of success for one surgery is 0.95, the probability of failure for one surgery is

P(none of the three are successful) = (0.05)(0.05)(0.05) 0.0001

EXAMPLE 3


3.) Find the probability that at least one of the three ACL surgeries is successful.

EXAMPLE 3 CONTINUED

The phrase “at least one” means one or more. The complement to the event “at least one is successful” is the event “none are successful.” use the complement to find the probability.

P(at least one is successful) =

USING THE MULTIPLICATION RULE

About 16,500 U.S. medical school seniors applied to residency programs in 2012. Ninety-five percent of the seniors were matched wit residency positions. Of those, 81.6% were matched with one of their top three choices. Medical students rank the residency programs in their order of preference, and program directors in the U.S. rank the students. The term “match” refers to the process whereby a student’s preference list and a program director’s preference list overlap, resulting in the placement of the student in a residency position.

T/

QUESTION 1

Find the probability that a randomly selected senior was matched with a residency position and it was one of the senior’s top three choices.

QUESTION 2

Find the probability that a randomly selected senior who was matched with a residency position did not get matched with one of the senior’s top three choices.

)

QUESTION 3

Would it be unusual for a randomly selected senior to be matched with a residency position and that it was one of the senior’s top three choices?

Similar to Question 1.

It is not unusual because the probability of a senior being matched with a residency position that was one of the senior’s top three choices is about 0.775. 77.5% is not usual. The unusual %’s are 3% or less and 97% or more.

TRY

http://www.regentsprep.org/regents/math/algebra/APR3/PracCond.htm

#’s 1 and 2



conditional probability 3.2 raise your hand when you are finished reading

Documents