11-3 Probability of Multiple Events

Quick Review

For any events A and B,

P ( A  or  B ) = P ( A ) + P ( B ) − P ( A  and  B ) . p open eh , or b close equals p open eh close plus p open b close minus p open eh , and b close .  When the occurrence of one event affects how a second event can occur, the events are dependent. When A and B are independent, P ( A  and  B ) = P ( A ) · P ( B ) . p open eh , and b close equals p open eh close middle dot p open b close .  For mutually exclusive events, P(A and B) = 0 so P(A or B) = P(A) + P(B).

Example

You roll a standard number cube. Are the events “roll a 1” and “roll an even number” mutually exclusive? Explain.

You cannot roll a 1 and an even number at the same time. The events are mutually exclusive.

Exercises

Classify each pair of events as dependent or independent.

20. A student in your algebra class is selected at random. One of the remaining students is selected at random.
21. You select a number 1 through 6 by tossing a standard number cube. You select a second number by tossing the number cube again.

Calculate each probability, given that P(A) = 0.3, P(B) = 0.7, and A and B are independent.

22. P(A and B)
23. P(A or B)

11-4 Conditional Probability

Quick Review

The probability that event B will occur, given that A has already occured, is the conditional probability P ( B | A ) = P ( A  and  B ) P ( A ) . p open b vertical line eh close equals . fraction p open eh , and b close , over p open eh close end fraction . .

Example

A standard number cube is rolled twice. If the first number rolled is a and the second is b, find P ( a  is even and  b > 2 )  and  P ( b  is even  |   b > 3 ) . p open eh . isevenand . b greater than 2 close , and p open b . iseven . vertical line b greater than 3 close .

Since number cube rolls are independent events,

P ( a  is even and  b > 2 ) = P ( a  is even ) ⋅ P ( b > 2 )   = 1 2 ⋅ 2 3 = 1 3 P ( b  is even |  b > 3 ) = P ( b > 3  and  b  is even ) P ( b > 3 )   = P ( 4  or  6 ) P ( 4  or  5  or  6 ) ⋅ 2 6 3 6 = 2 3 . table with 4 rows and 2 columns , row1 column 1 , p open eh . isevenand . b greater than 2 close , column 2 equals p open eh , iseven , close dot p open b greater than 2 close , row2 column 1 , , column 2 equals , 1 half , dot , 2 thirds , equals , 1 third , row3 column 1 , p open b . isevenvertical line . b greater than 3 close , column 2 equals . fraction p open b greater than 3 , and , b , iseven , close , over p open b greater than 3 close end fraction , row4 column 1 , , column 2 equals . fraction p open 4 , or , 6 close , over p open 4 , or , 5 , or , 6 close end fraction . dot . fraction 2 sixths , over 3 sixths end fraction . equals , 2 thirds . . , end table

Exercises

Calculate each probability, given that P(A) = 0.3, P(B) = 0.7, and A and B are independent.

24. P ( A   |   B ) p open eh vertical line b close
25. P ( B   |   A ) p open b vertical line eh close

Calculate each probability, given that P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.1.

26. P ( A   |   B ) p open eh vertical line b close
27. P ( B   |   A ) p open b vertical line eh close
28. P ( A  and  B   |   A  or  B ) p open eh , and b vertical line eh , or b close

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