3.8: Review Exercises
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According to the results of a survey, 200 people said they prefer cats, 160 said they prefer dogs, while 40 said they don’t like either pet.
- Find the empirical probability that a person prefers dogs. Give the probability as a fraction in simplest form, as a decimal, and as a percent.
- Based on this probability, how many people should prefer dogs if a survey of 500 people is taken?
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A six-sided die (with sides numbered 1 through 6) is tossed. Find the probability of each event as a simplified fraction.
- getting an even number.
- getting a number less than 5.
- getting a 7.
- A ball is drawn randomly from a jar that contains 6 red balls, 2 white balls, and 5 yellow balls. Find the probability of each event as a simplified fraction.
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This set of 15 cards is used in a game. The cards are well-shuffled, and then you pick one card. Calculate each of the following.
- P(selecting a circle)
- P(selecting a black shape)
- P(selecting a black heart)
- P(selecting a triangle or a star)
- P(selecting a heart or a black shape)
- P(selecting a card that is not a star)
- P(selecting a circle given the shape is white)
- P(selecting a white shape given it is a circle)
- odds in favor of selecting a triangle
- odds against selecting a black shape
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The sample space for rolling two dice is shown as ordered pairs. Use the sample space to find the probability of each event.
- P(roll the same value on both dice)
- P(roll a 5 on at least one of the two dice )
- P(sum of the dice is 8)
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When a button is pressed, a computer program outputs a random odd number greater than 1 but less than 9. Consider the experiment "press the button twice."
- Write the sample space for this experiment. Use an ordered pair for each outcome.
- List the outcomes in the event \(A =\) "the sum of the two numbers is 10 or more. " Then find \(P(A)\).
- List the outcomes in the event \(B =\) " both numbers are the same ." Then find \(P(B)\).
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A box contains one green, one red, one yellow, and one blue marble. You will select one marble and record its color. Without replacing the marble, you will select another marble and record its color.
- Write the sample space for this experiment. Use an ordered pair for each outcome.
- What is the probability that you get a red or yellow marble?
- What is the probability that you won't get a yellow marble?
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An urn contains 10 red balls, 15 white balls, and 20 black balls. A single ball is selected at random. Find these odds written as ratios in lowest terms.
- odds in favor of selecting a red ball
- odds against drawing a white ball.
- The odds in favor of Julio winning the game are \(9:14\) What is the probability that Julio wins?
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The probability your English professor is in their office is \(\frac{3}{5}\).
- What is the probability that your English professor is not in their office?
- What are the odds in favor of your English professor being in their office?
- There is a 35% probability that the bus is late. What are the odds in favor of the bus being late?
- For a certain tutor the probabilities for various numbers of student no-shows per day are shown in the following table. What is the expected value for the number of no-shows per day?
| Number of No Shows | 0 | 1 | 2 | 3 | 4 | ||
|---|---|---|---|---|---|---|---|
| Probability | 0.40 | 0.25 | 0.20 | 0.10 | 0.05 |
- An insurance company insures a house worth $250,000 for an annual premium of $500. If the probability of the house being destroyed is 0.0015 and assuming either total loss or no loss, what is the insurance company’s expected annual profit for the policy?
- A friend devises a game that is played by rolling a single six-sided die once. If you roll a 6, he pays you $3; if you roll a 5, he pays you nothing; if you roll a number less than 5, you pay him $1. What is your expected gain or loss for playing this game. Is this a fair game?
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A student believes he has an 80% chance of passing his English class, 30% chance of passing his math class, and a 25% chance of passing both classes.
- What is the probability he will not pass his English class?
- What is the probability he passes English or math?
- What is the probability he passes neither class?
- What is the probability of passing math given he passed English?
- A jar contains 6 red marbles numbered 1 to 6 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar. Find the probability the marble is red or odd-numbered.
- On any given night, the probability that Nick has a cookie for dessert is 15%. The probability that Nick has ice cream for dessert is 50%. The probability that Nick has a cookie and ice cream is 10%. What is the probability that Nick has a cookie or ice cream for dessert?
- Suppose \(P(B) = 0.68\), \(P(A \text{ or } B) = 0.90\), and \(P(A \text{ and } B) = 0.45\). Find \(P(A)\).
- The table below shows the number of credit cards owned by a group of individuals. One person will be selected at random. Find the probability of each event below.
\(\begin{array}{|l|l|l|l|l|}
\hline & \text { Zero } & \text { One } & \text { Two or more } & \text { Total } \\
\hline \text { Male } & 9 & 5 & 19 & 33 \\
\hline \text { Female } & 18 & 10 & 20 & 48 \\
\hline \text { Total } & 27 & 15 & 39 & 81 \\
\hline
\end{array}\)
- the person is female
- the person has exactly 0 or 1 credit card
- the person is male and has exactly one credit card
- the person is male or has no credit cards.
- the person is female given that they have 2 or more credit cards
- the person has 0 credit cards if they are male
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Bert has a well-shuffled standard deck of 52 cards, from which he draws one card. Ernie has a 12-sided die, which he rolls at the same time Bert draws a card. Compute the probability that
- Bert gets a Jack and Ernie rolls a five.
- Bert gets a heart and Ernie rolls a number less than six.
- Bert gets a face card (Jack, Queen or King) and Ernie rolls an even number.
- Bert gets a card that is not a Jack and Ernie rolls a number that is not twelve.
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A multiple-choice test consists of 4 questions. Each question has answer choices
a
through
e
so the chance of guessing the correct answer is \(\frac{1}{5}=0.2.\) Suppose a student guesses on all 4 questions.
- What is the probability the student guesses all the answers correctly?
- What is the probability the student guesses none of the answers correctly?
- What is the probability the student guesses at least one of the answers correctly?
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A jar contains 6 red balls, 2 white balls, and 5 yellow balls. Two balls are randomly selected
with replacement
. Find the probability of each event.
- both balls are red.
- neither of the balls are red.
- first ball is red and second ball is white.
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A jar contains 6 red balls, 2 white balls, and 5 yellow balls. Two balls are randomly selected
without replacement
. Find the probability of each event.
- both balls are red.
- neither of the balls are red.
- first ball is red and second ball is white.
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A math class consists of 25 students, 14 female and 11 male. Three different students are selected at random to participate in a probability experiment. Compute the probability that
- a male is selected, then two females.
- a female is selected, then two males.
- three males are selected.
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Suppose that 21% of people own dogs. You pick three people at random. Assuming independence between the people, find each probability:
- all 3 people own a dog.
- none of the 3 people own a dog.
- at least one of the 3 people own a dog.
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A jar contains 5 red marbles numbered 1 to 5 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar. Find the probability the marble is
- even-numbered given that the marble is red.
- red given that the marble is even-numbered.
- The probability that Jack oversleeps is 40%.The probability Jack oversleeps and is late for work is 10%. What is the probability that Jack is late for work given he oversleeps?
- A boy owns 2 pairs of pants, 3 shirts, 8 ties, and 2 jackets. How many different outfits can he wear to school if he must wear one of each item?
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A computer password must be 6 characters long. How many passwords are possible if
- only the 26 lowercase letters of the alphabet are allowed and letters may be repeated?
- only the 26 lowercase letters of the alphabet are allowed and letters may not be repeated?
- both lowercase and uppercase letters are allowed and letters may be repeated?
- A doctor needs to schedule 4 patients for a vaccine at different times. In how many ways can she she schedule the 4 patients?
- In how many ways can first, second, and third prizes be awarded in a contest with 50 contestants?
- In how many ways can you select 4 different pizza toppings from 12 available toppings on the menu?
- At a baby shower 15 guests are in attendance and 5 of them are randomly selected to receive identical door prizes. How many ways can the prizes be awarded?
- In a lottery game, a player picks six different numbers from 1 to 50. How many different choices of numbers does the player have for his lottery ticket?
- A congressional committee consists of 10 members: 6 Democrats and 4 Republicans. A sub-committee of 5 members must be formed containing 3 Democrats and 2 Republicans. How many ways can the sub-committee be chosen?
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A gardener has 12 identical-looking tulip bulbs, of which 7 will produce yellow tulips and 5 will be red. She randomly chooses 4 of the bulbs and plants them.
- What's the probability that all 4 tulips will be yellow when they bloom?
- What's the probability that 2 of the tulips will be yellow and 2 of the tulips will be red when they bloom?
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Find the probability of being dealt each of the following 5-card poker hands:
- all hearts.
- exactly three red cards.