# theoretical probability

1. The theoretical probability of undesirable side effects resulting from taking Pyrene is 1 in 15. If 60 people take Pyrene to relieve congestion, theoretically how many will encounter undesirable side effects?

2. A bag contains 16 yellow marbles, 6 green marbles, and 7 red marbles. What is the chance of drawing a yellow marble? If a yellow marble is drawn the first time and then a second marble is drawn without replacement, what is the probability of drawing a green marble? Give solutions exactly in reduced fraction form, separated by a comma.

3. The odds in favor of Sprinting Sarah winning the horse race are 19:3. Determine the probability that Sprinting Sarah wins the horse race.

3/19

22/3

3/22

19/22

19/3

4. The odds against Trotting Carlos winning the horse race are 29:7. Determine the probability that Trotting Carlos wins the horse race.

7/29

36/7

29/36

29/7

7/36

5. A mini license plate for a toy car must consist of a letter followed by two numbers. Each letter must be a C, A or R. Each number must be a 3 or 7. Repetition of digits is permitted.

1. Use the counting principle to determine the number of points in the sample space.

2. Construct a tree diagram to represent this situation List the sample space.

3. List the sample space.

4. Determine the exact probability of creating a mini license plate with a 7. Give solution exactly in reduced fraction form

6. Suppose 42 newly created pairs of jeans are placed in a bin to be checked for quality. Of these, 4 will not be of high enough quality to sell. Suppose that two pairs of jeans are randomly selected, without replacement, from the bin. What is the chance that both pairs of jeans selected will be substandard? Show step by step work! Give solution as a reduced fraction in lowest terms and approximated to the nearest ten thousandth.

7. The FM Radio stations with high signal strength in New Orleans break down into the following categories:

Hip hop 2

Religious 6

Country 2

Rock 3

Talk Radio 2

Other 5

If someone driving in New Orleans tuned into an FM radio station with high signal strength at random, what would be the chances he would not get a Country station? Give answer as a fraction in lowest terms.

8. The pets a particular veterinarian saw yesterday break down into the following categories:

Cats 10

Dogs 18

Bird 4

Ferret 3

Lizard 1

If there is one folder per pet and Valerie selected a folder at random from yesterday's stack of folders, what would be the chances she would not get a bird's chart? Give answer as a fraction in lowest terms

9. If casino Keno were fair, how much should it pay to a player who bets $10 to bet 6 numbers and matches 5 numbers? The odds against matching 5 numbers when you bet 6 numbers in casino Keno are about 322 to 1.

1. Of 31 college seniors at Southern Swampland University, 17 preferred pepperoni pizza, 12 preferred supreme, and 2 preferred cheese. If we picked a college senior at Southern Swampland University at random, theoretically what is the probability that he or she would prefer pepperoni? Give solution exactly in reduced fraction form.

2. A bag contains 6 purple marbles, 7 azure marbles and 10 orange marbles. What is the chance of drawing a purple marble? If an purple marble is drawn then placed back into the bag, and a second marble is drawn, what is the probability of drawing an orange marble? Give solutions exactly in reduced fraction form, separated by a comma

3. A boxed DVD set contains 18 movies from a certain studio. Ten of them are comedies, five are thrillers, and three are dramas. What is the probability that one DVD chosen from the box at random is a thriller or a comedy? Show step by step work! Give answer as a fraction in lowest terms.

4. According to the U. S. Census Bureau, the total 2008 U.S. population was 303,824,640. The chart below summarizes the 2008 population for five U.S. States.

State 2008 Population

Missouri 5,911,605

Pennsylvania 12,448,279

Tennessee 6,214,888

Utah 2,736,424

Washington 6,549,224

SOURCE: U. S. Census Bureau

What is the probability that a randomly selected U.S. resident did not live in Utah? Show step by step work. Round solution to the nearest thousandth.

5 .

A mini license plate for a toy car must consist of a letter followed by two numbers. Each letter must be an R, U or N. Each number must be a 2 or 6. Repetition of digits is permitted.

1. Use the counting principle to determine the number of points in the sample space.

2. Construct a tree diagram to represent this situation

3. List the sample space.

4. Determine the exact probability of creating a mini license plate with an R. Give solution exactly in reduced fraction form.

6. A parent can choose from 7 types of protein, 4 vegetables and 5 desserts. If the parent serves a meal of 1 protein, 1 vegetable and 1 dessert to the family, how many different meals can be served?

7. A card is selected from a standard deck of 52 playing cards. Find the probability of selecting

1. a prime number under 10 given the card is red. (1 is not prime.)

2. a King, given that the card is not a heart.

3. a nine given the card is a face card. (An ace is not a face card.)

Please show step by step work. Give all solutions exactly in reduced fraction form.

8. Last fall, a gardener planted 55 iris bulbs. She found that only 45 of the bulbs bloomed in the spring.

1. Find the empirical probability that an iris bulb of this type will bloom. Give answer as a fraction in lowest terms.

2. How many of the bulbs should she plant next fall if she would like at least 51 to bloom?

9. In how many ways can 8 instructors be assigned to five sections of a course in mathematics?

10.

Which pair has equally likely outcomes? List the letters of the two choices below which have equal probabilities of success, separated by a comma.

A. drawing a black seven out of a standard 52 card deck given it's not a face card or an ace.

B. rolling a total of 4 on two fair six sided dice

C. drawing a three out of a standard 52 card deck given it's not a face card or an ace.

D. rolling a total of 9 on two fair six sided dice

E. rolling a total of 7 on two fair six sided dice

11. In how many different ways can the top eight new indie bands be ranked on a top eight list? The top hit song for each of the eight bands will compete to receive monetary awards of $1000, $500, $250 and $100, respectively. In how many ways can the awards be given out?

12. How many different ways are there for an admissions officer to select a group of 5 college candidates from a group of 9 applicants for an interview?

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#### Solution Preview

1. The theoretical probability of undesirable side effects resulting from taking Pyrene is 1 in 15. If 60 people take Pyrene to relieve congestion, theoretically how many will encounter undesirable side effects?

Think of this as a proportion problem. If 1 in 15 get the side effects, then how many in 60 will get them. This is the proportion 1 / 15 = x / 60. You can solve for x by multiplying both sides by 60 to get 60 / 15 = x or x = 3. The ratios 1/15 and 3/60 are equal so the answer is 3.

2. A bag contains 16 yellow marbles, 6 green marbles, and 7 red marbles. What is the chance of drawing a yellow marble? If a yellow marble is drawn the first time and then a second marble is drawn without replacement, what is the probability of drawing a green marble? Give solutions exactly in reduced fraction form, separated by a comma.

There are a total of 29 marbles and 16 of them are yellow, so the chance of drawing a yellow is 16 / 29. After one yellow marble is drawn, there are only 28 marbles left since the yellow was not replaced. Six of those are green. So this gives 6 / 28. The probability of first drawing a yellow and then getting a green is found by multiplying the two probabilities together to get 16 / 29 * 6 / 28 = 96 / 812 or 24 / 203 after dividing the both numbers by 4. So the answers are 16 / 29, and 24 / 203.

3. The odds in favor of Sprinting Sarah winning the horse race are 19:3. Determine the probability that Sprinting Sarah wins the horse race.

3/19

22/3

3/22

19/22

19/3

The odds are a ratio of the unfavorable outcomes to the favorable outcomes. That means that out of a total of 22 times, it is expected that Sprinting Sarah will lose 19 times and win 3 times. Writing this as a probability is 3 / 22 since he will win a total of 3 out of 22 races.

4. The odds against Trotting Carlos winning the horse race are 29:7. Determine the probability that Trotting Carlos wins the horse race.

7/29

36/7

29/36

29/7

7/36

The odds are that out of a total of 36 races, Trotting Carlos will likely win 7 times but lose 29 times. So the probability that he will win will be 7 / 36.

5. A mini license plate for a toy car must consist of a letter followed by two numbers. Each letter must be a C, A or R. Each number must be a 3 or 7. Repetition of digits is permitted.

1. Use the counting principle to determine the number of points in the sample space.

2. Construct a tree diagram to represent this situation List the sample space.

3. List the sample space.

4. Determine the exact probability of creating a mini license plate with a 7. Give solution exactly in reduced fraction form

1. There are a total of three letters possible and four different number combinations (33, 37, 73, 77) that can be used in the license plate. So the counting rule gives 3 * 4 = 12 possible different license plates.

2.

C A R

3 7 3 7 3 7

3 7 3 7 3 7 3 7 3 7 3 7

3. Sample Space = {C33, C37, C73, C77, A33, A37, A73, A77, R33, R37, R73, R77}

4. There are a total of 12 different plates with 9 of them having a 7, so the probability is 9 / 12 or 3 / 4 since both 9 and 12 are divisible by 3. So the answer is 3 / 4.

6. Suppose 42 newly created pairs of jeans are placed in a bin to be checked for quality. Of these, 4 will not be of high enough ...

#### Solution Summary

The theoretical probability is discussed.