12-7 Theoretical and Experimental Probability

Quick Review

An event is an outcome or group of outcomes. The probability of an event, which indicates how likely it is to occur, is written P(event). When all possible outcomes are equally likely, the theoretical probability of an event is given by P ( event ) = number of favorable outcomes number of possible outcomes . p open , event , close equals . fraction numberoffavorableoutcomes , over numberofpossibleoutcomes end fraction . .

Example

What is the theoretical probability that a randomly chosen date is a day beginning with a T?

There are 2 favorable outcomes, Tuesday and Thursday.

There are 7 possible outcomes, the 7 days of the week.

P ( day beginning with a T ) = number of favorable outcomes number of possible outcomes   = 2 7 table with 2 rows and 2 columns , row1 column 1 , p open . daybeginningwithacap t . close , column 2 equals . fraction numberoffavorableoutcomes , over numberofpossibleoutcomes end fraction , row2 column 1 , , column 2 equals , 2 sevenths , end table

The probability of a day beginning with a T is 2 7 . 2 sevenths , .

Exercises

The spinner below is divided into six equal sections. Find the theoretical probability of landing on the given sections of the spinner.

37. P(even)
38. P(odd)
39. P(5)
40. P(not 3)
41. P(7)
42. P(more than 4)
43. Apples An apple farmer finds that he has to throw out 15 bad apples from the 225 he has picked. What is the experimental probability that the next apple he picks will be good?

12-8 Probability of Compound Events

Quick Review

You can use a formula to find the probability of a compound event involving two events A and B.

Mutually exclusive events: P ( A  or  B ) = P ( A ) + P ( B ) p open eh , or b close equals p open eh close plus p open b close

Overlapping events:

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

Independent events: 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

Dependent events: P ( A  then  B ) = P ( A ) · P ( B  after  A ) p open eh , then b close equals p open eh close middle dot p open b , after eh close

Example

You roll a number cube and flip a coin. What is the probability that you roll a 5 and the coin comes up heads?

Rolling a 5 and flipping heads are independent events.

P ( 5  and heads  ) = P ( 5 ) · P ( heads ) = 1 6 ⋅ 1 2 = 1 12 p open 5 . andheads . close equals p open 5 close middle dot . p . open , heads , close . equals , 1 sixth , dot , 1 half , equals , 1 twelfth

The probability of rolling a 5 and flipping heads is 1 12 . 1 twelfth , .

Exercises

You randomly pick two marbles from a bag containing 3 yellow marbles and 4 red marbles. You pick the second marble without replacing the first marble. Find each probability.

44. P(red then red)
45. P(yellow then red)

You roll a number cube twice. Find each probability.

46. P(6 then 3)
47. P(odd then even)

Are the two events dependent or independent? Explain.

48. You pick one of 7 names from a hat and then pick a second name without replacing the first one.
49. You spin a spinner with 5 equal sections and pick a marble from a bag containing 2 green marbles and 4 blue marbles.

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