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Algebra II & Trigonometry Theoretical Probability

**A2.S.13 Calculate theoretical probabilities, including geometric applications**

## Theoretical Probability

Probability is the likelihood that a given event will occur. Probability =

Some properties of probability:

o Probability is expressed as a fraction or a decimal between 0 and 1.

o If the probability of an event is 1, the event is certain to occur.

o The probability of an impossible event is 0.

o The sum of all probabilities in a situation equals 1.

o The probability of an event E occurring is written P(E).

Ex: A box of golf balls contains 3 yellow, 4 white, and 1 red. If Ryan chooses one golf ball from the box at random, find the probability that the golf ball is yellow.

Solution: There are 3 yellow golf balls out of 8 total balls, so the probability of choosing a yellow ball is .

### Probability of More than One Event

P(A and B) =

If the events are independent, we won’t have to worry about one event affecting another.

Ex: The spinner to the right is spun twice, find the probability of P(red and then blue).

Solution: The first spin is independent of the second spin.

One can see from the spinner that P(red) = and P(blue) = .

Therefore the P(red and then blue) =

There are problems in which events are dependent. These situations usually involve events involving no replacement.

Ex: Suppose a box contains 5 math textbooks and 4 science text books. What is the probability of picking out 2 math books in a row.

Solution: P(math and math) =

Notice that the probability of success changes.

### Probability of At Least One Event

Ex: A single card is randomly selected from a standard deck of 52 playing cards. Find the probability that the chosen card is i. an ace or a king. ii. A black card or a 10

Ex: Ms. Rito needs two seniors to head the November blood drive. If 12 students volunteer, 4 boys and 8 girls, what is the probability that one boy and one girl will be chosen to head the blood drive?

Solution: Find the number of ways 1 boy and 1 girl can be selected from among the 12 volunteers and divide by the total number of ways 2 people can be selected.

It is not necessary to put the fraction in lowest terms.

1. A fair coin is tossed and an unbiased die is thrown. What is the probability that the results are heads, 5?

(1) (2) (3) (4)

2. In playing the game Yahtzee, 5 dice are tossed simultaneously. What is the probability that all 5 dice show 2?

(1) (2) (3) (4)

3. Jake has eight crayons of different colors: pale blue, sky blue, navy blue, fire engine red, cherry red, forest green, lime green, and black. If he chooses two crayons without looking, what is the probability that both are blue?

(1) (2) (3) (4)

4. Twelve students are in the Saturday morning Driver Education class, 7 boys and 5 girls. If there are 3 students assigned to each car, what is the probability that Maria, Grace, and Timothy are all assigned to Mr. Krieg’s car?

(1) (2) (3) (4)

5. There are eleven sections of Freshman Composition at Oneida Community College and three sections of International Diplomacy. If Kai has his classes scheduled first, what is the probability that Andrew is scheduled in the same sections?

(1) (2) (3) (4)

6. If the probability of an event’s occurring is , what is the probability of the event not occurring?

(1) (2) (3) (4)

7. From a group of 8 Democrats and 6 Republicans, a 5-member committee needs to be formed. Which of the following represents the probability that the committee contains 3 Democrats and 2 Republicans?

(1) (2) (3) (4)

8. The school musical next year will be Grease. After initial tryouts, seven girls are still in competition for the parts of Sandy, Frenchy, and Rizzo. If the selection is made by drawing names, what is the probability that Savannah, Loghan, and Kathleen are chosen for the roles of Sandy, Frenchy, and Rizzo in that order?

(1) (2) (3) (4)

9. In horse racing, a trifecta is a bet in which the first, second, and third horses are predicted in that precise order. If George bets a trifecta in a race in which 9 horses are running, what is the probability he might win, assuming that all of the horses have equal talent?

(1) (2) (3) (4)

10. A carton of books contains 5 different mysteries, 3 different horror novels, and 2 different romance novels. If Judith picks 3 books at random from the carton, what is the probability that she has selected one from each category?

(1) (2) (3) (4)