Q: A bell tower has 12 bells, each with a different note. Once a week, 6 bells are rung in sequence. No bell is rung more than once. How many possible unique sequences are there?

$$12 \times 11 \times 10 \times 9 \times 8 \times 7 = 665280$$

Q: One option in a roulette game is to bet on red. There are 18 red compartments, 18 black compartments, and two compartments that are neither red nor black. If the ball lands on red, you get to keep the $$10 that you paid to play the game and you are awarded 10. If $$the ball lands elsewhere, you are awarded nothing and the $$10 that you bet is collected. What is the expected value for playing roulette if you bet 10 on $$red?

$$\dfrac{18}{38} \times 10 + \dfrac{20}{38} \times (-10) = -\0.53$

Question 1

A person can order a new car with a choice of 8 possible colors, with or without air conditioning, with or without heated seats, with or without anti-lock brakes, with or without power windows, and with or without a CD player. In how many different ways can a new car be ordered in terms of these options?

$$	ext{(Hint: Use the multiplication principle)}$$

Accepted Answer

$$8 	imes 2^5$ = 256$$

Question 2

Evaluate the factorial expression: $$\dfrac{322!}{320!}$$

Accepted Answer

$$322 	imes 321$$

Question 3

Evaluate the combination: $$\displaystyle {21 \choose 7}$$

Accepted Answer

$$116280$$

Question 4

A bell tower has 12 bells, each with a different note. Once a week, 6 bells are rung in sequence. No bell is rung more than once. How many possible unique sequences are there?

Accepted Answer

$$12 	imes 11 	imes 10 	imes 9 	imes 8 	imes 7 = 665280$$

Question 5

To win at LOTTO in a certain state, one must correctly select 5 numbers from a collection of 50 numbers (one through 50). The order in which the selection is made does not matter. How many different selections are possible?

Accepted Answer

$$\displaystyle {50 \choose 5} = 2118760$$

Question 6

Suppose a fair, six-sided die is rolled. What is the probability of rolling a prime number?

Accepted Answer

$$\dfrac{3}{6} = \dfrac{1}{2}$$

Question 7

Suppose a fair, six-sided die is rolled. What is the probability of rolling a number more than 4?

Accepted Answer

$$\dfrac{2}{6} = \dfrac{1}{3}$$

Question 8

One option in a roulette game is to bet on red. There are 18 red compartments, 18 black compartments, and two compartments that are neither red nor black. If the ball lands on red, you get to keep the $$10 that you paid to play the game and you are awarded 10. If $$the ball lands elsewhere, you are awarded nothing and the $$10 that you bet is collected. What is the expected value for playing roulette if you bet 10 on $$red?

Accepted Answer

$$\dfrac{18}{38} 	imes 10 + \dfrac{20}{38} 	imes (-10) = -\0.53$

Question 9

A box contains 40 widgets, 5 of which are defective. If 5 are sold at random, what is the probability that all are defective?

Accepted Answer

$$\dfrac{{5 \choose 5}}{{40 \choose 5}}$$

Question 10

A box contains 40 widgets, 5 of which are defective. If 5 are sold at random, what is the probability that none are defective?

Accepted Answer

$$\dfrac{{35 \choose 5}}{{40 \choose 5}}$$

Question 11

Given the following data for people in Country X who moved in a recent year (in millions):
Owner: 16.2, Renter: 25.8, Total: 42.0. What is the probability that a randomly selected person from Country X who moved was an owner?

Accepted Answer

$$\dfrac{16.2}{42.0}$$

Question 12

Given the same data, what is the probability that a randomly selected person from Country X who moved was a renter or they moved to a different continent?

Accepted Answer

$$\dfrac{25.8 + 1.1 - 0.9}{42.0}$$

Question 13

Given the same data, what is the probability that a randomly selected person from Country X who moved was an owner given that they moved to a different country on the same continent?

Accepted Answer

$$\dfrac{3.6}{9.4}$$

Question 14

Suppose you draw one card from a 52-card deck. What is the probability that it is not a face card?

Accepted Answer

$$\dfrac{40}{52} = \dfrac{10}{13}$$

Question 15

Suppose you draw one card from a 52-card deck. What is the probability that it is either a club or a black card?

Accepted Answer

$$\dfrac{26}{52} = \dfrac{1}{2}$$

Question 16

The winner of a raffle will receive a new car. If 25,000 raffle tickets were sold and you purchased 20 of those tickets, what are the odds against your winning the car?

Accepted Answer

$$\dfrac{24980}{20} = 1249:1$$

Question 17

There are 38 chocolates in a box: 12 with nuts, 12 with caramel, and 14 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. What is the probability of selecting 2 solid chocolates in a row?

Accepted Answer

$$\dfrac{14}{38} 	imes \dfrac{13}{37}$$

Question 18

An unfair coin is flipped repeatedly. The probability of getting heads for this coin is 0.48. What is the probability of getting heads two flips in a row?

Accepted Answer

$$0.48 	imes 0.48 = 0.2304$$

Question 19

An unfair coin is flipped repeatedly. The probability of getting heads for this coin is 0.48. What is the probability of getting tails for four flips in a row?

Accepted Answer

$$0.52^4 = 0.0730$

Question 20

An unfair coin is flipped repeatedly. The probability of getting heads for this coin is 0.48. What is the probability of a tail at least once in ten flips?

Accepted Answer

$$1 - 0.48$$^{10}$$ \approx 0.9994$$

Question 21

Based on the tuberculosis (TB) test data: 11 people had a positive screening test and TB, 6 had a positive screening test and no TB, 121 had a negative screening test and no TB. What is the probability that a randomly selected person from this group has TB?

Accepted Answer

$$\dfrac{11}{138}$$

Question 22

Suppose a person from the TB test group has a positive test screening. What is the probability that they have TB?

Accepted Answer

$$\dfrac{11}{17}$$

Liberal Arts Math Exam 3 Study Guide: Probability, Counting, and Combinatorics

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