Question 1

Use the at least once rule to find the indicated probability.
-Find the probability of getting at least one tail when tossing 6 fair coins.

Accepted Answer

The probability of getting at least one tail when tossing 6 fair coins is found by subtracting the probability of getting no tails (all heads) from 1. The probability of getting all heads in one toss is 1/2, so for 6 tosses it's (1/2)^6 = 1/64. Therefore, the probability of getting at least one tail is 1 - 1/64 = 63/64 = 0.984.

Question 2

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-Three fair coins are tossed. Find the probability of getting the same thing on all three coins.

Accepted Answer

When three fair coins are tossed, the possible outcomes are HHH, HHT, HTH, THH, TTH, THT, HTT, and TTT. Out of these 8 outcomes, only HHH and TTT have the same result on all three coins. Thus, the probability is 2/8 = 1/4.

Question 3

Use the at least once rule to find the indicated probability.
-In a blood testing procedure, blood samples from 5 people are combined into one mixture. The mixture will only test negative if all the individual samples are negative. If the probability that an
Individual sample tests positive is 0.08, what is the probability that the mixture will test positive?

Accepted Answer

The probability that an individual sample tests negative is 0.92 (1 - 0.08). The probability that all 5 samples test negative is 0.92^5. The probability that at least one sample tests positive (and thus the mixture tests positive) is 1 - 0.92^5 = 0.341.

Question 4

Find the expected value.
-In a large casino, the house wins on one of its games with a probability of 50.9%. All bets are 1 : 1 . If you win, you gain the amount you bet; if you lose, you lose the amount you bet. If you play 140
Games in an evening, betting $1 each time, how much should you expect to win or lose?

Accepted Answer

The expected value can be calculated by multiplying the outcomes by their probabilities and summing these products. Here, the probability of winning is 49.1% (100% - 50.9%), and the probability of losing is 50.9%. For each game, the expected value is (0.491 * $1) + (-0.509 * $1) = -$0.018. Over 140 games, the expected value is 140 * -$0.018 = -$2.52, meaning you should expect to lose $2.52.

Question 5

Solve the problem.
-If it has been determined that the probability of an earthquake occurring on a certain day in a certain area is 0.05, what are the odds against an earthquake?

Accepted Answer

The odds against an event are calculated as the ratio of the probability of the event not happening to the probability of the event happening. Here, the probability of not having an earthquake is 1 - 0.05 = 0.95. Thus, the odds against an earthquake are 0.95 to 0.05, which simplifies to 19 to 1.

Question 6

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-A class consists of 11 women and 44 men. If a student is randomly selected, what is the probability that the student is a woman?

Accepted Answer

The answer of Find the indicated probability. Round your answer...

Question 7

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-One card is selected from a deck of cards. Find the probability of selecting a heart or a card less than 8 . (Note: The ace is considered a low card.)

Accepted Answer

The answer of Find the indicated probability. Round your answer...

Question 8

Use the relative frequency method to estimate the probability. Round your answer to 2 decimal places when necessary.
-Of the last 100 people who failed the lie detector test, 23 turned out to be telling the truth. What is the probability that the next person who fails the test is actually telling the truth?

Accepted Answer

The probability is calculated by dividing the number of people who were telling the truth (23) by the total number of people who failed the test (100), which equals 0.23.

Question 9

Determine whether the events A and B are independent.
-12 jurors are selected from a pool of 20 Event 
A: The first person selected is a woman
Event 
B: The second person selected is a woman

Accepted Answer

The selection of the first person affects the composition of the pool from which the second person is selected, thus altering the probability of the second event based on the outcome of the first.

Question 10

Solve the problem. Round your answer to 2 decimal places when necessary.
-A U.S. city with a population of 2.2 million reports 30,724 births in one year. How many people are born per day in the city?

Accepted Answer

To find the number of births per day, divide the total number of births in a year by the number of days in a year: 30,724 births / 365 days = 84.17 births per day, rounded to 84 births per day.

Question 11

Decide whether events A and B are overlapping or non-overlapping.
-Event A is that you study at least 5 hours for your math test. Event B is that you study at most 6 hours for your math test.

Accepted Answer

Events A and B are overlapping because there is a range of time (from 5 to 6 hours) where both events can occur simultaneously.

Question 12

Solve the problem.
-The library is to be given 5 books as a gift. The books will be selected from a list of 21 titles. If each book selected must have a different title, how many possible selections are there?

Accepted Answer

This is a combination problem, not a permutation, because the order in which the books are selected does not matter. The formula for combinations is C(n, k) = n! / [k!(n-k)!], where n is the total number of items to choose from, k is the number of items to choose, and "!" denotes factorial, the product of all positive integers up to that number. Here, n=21 (the total number of books) and k=5 (the number of books to select). Thus, C(21, 5) = 21! / [5!(21-5)!] = 20,349.

Question 13

Solve the problem. Round your answer to 2 decimal places when necessary.
-The overall U.S. death rate for 60-62 year-olds is approximately 13 deaths per 1000 people. If there are about 12,142 60-62 year olds in a U.S. city, how many 60-62 year olds can be expected to die in
A year in that city?

Accepted Answer

To find the expected number of deaths, multiply the death rate by the population: (13/1000) * 12,142 = 157.846, which rounds to 158 deaths.

Question 14

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-A bag contains 6 red marbles, 3 blue marbles, and 7 green marbles. If a marble is randomly selected from the bag, what is the probability that it is blue?

Accepted Answer

The total number of marbles is 6 (red) + 3 (blue) + 7 (green) = 16. The probability of selecting a blue marble is the number of blue marbles (3) divided by the total number of marbles (16), which is $\frac{3}{16}$.

Question 15

Make a probability distribution for the given set of events.
-Suppose that you start with a normal deck of 52 cards and remove everything except the twos, threes, and fours. So you now have a deck of twelve cards consisting of four twos, four threes, and
Four fours. A card is drawn at random and replaced then another card is drawn at random and
Replaced. Make a probability distribution for the sum of the two numbers that are drawn.

Accepted Answer

The sums and their probabilities are calculated based on the combinations of drawing two cards (with replacement) from a deck consisting of only twos, threes, and fours. The possible sums are 4 (two twos), 5 (a two and a three), 6 (two threes, a two and a four, or two fours), 7 (a three and a four), and 8 (two fours). The probabilities are determined by the number of ways each sum can occur divided by the total number of possible outcomes (9, since each card draw has 3 outcomes and there are 2 draws: 3x3=9).

Question 16

Solve the problem.
-License plates are made using 2 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed?

Accepted Answer

There are 26 choices for each letter and 10 choices for each digit. Therefore, the total number of plates is 26 * 26 * 10 * 10 * 10 = 676,000.

Question 17

Solve the problem. Round your answer to 2 decimal places when necessary.
-The population of a certain country is about 132 million. The overall birth rate is 14.3 births per 1000. Approximately how many births were there in the country?

Accepted Answer

To find the number of births, multiply the population by the birth rate per 1000. So, $132,000,000 \times \frac{14.3}{1000} = 1,887,600$ births, which rounds to approximately 1.89 million births.

Question 18

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-Two cards are selected without replacement from a standard deck of 52 cards. What is the probability that both cards are the same color (i.e.either both black or both red)?

Accepted Answer

The probability of drawing two cards of the same color from a standard deck without replacement can be calculated by considering the two cases separately (both black or both red) and then adding those probabilities together. For the first card, the probability doesn't matter because any card can start the process. For the second card, however, the probability that it matches the color of the first card is 25/51 because one card of that color has already been removed (26 cards of the same color initially, minus the one drawn). So, the calculation is (26/52) * (25/51) for one color, and you double it to include both colors: 2 * (26/52) * (25/51) = 0.490196, which rounds to 0.490.

Question 19

Decide whether events A and B are overlapping or non-overlapping.
-A student is selected at random. Event A is that the student speaks Spanish and event B is that the student speaks Chinese.

Accepted Answer

Events A and B are overlapping because it is possible for a student to speak both Spanish and Chinese, meaning the two events can occur at the same time.

Question 20

Find the indicated probability.
-A die is rolled 50 times with the following results. $$\begin{array} { l | r r r r r r } 
\text { Outcome } & 1 & 2 & 3 & 4 & 5 & 6 \
\hline \text { Frequency } & 7 & 4 & 13 & 23 & 0 & 3
\end{array}$$ Compute the empirical probability that the die comes up a 5.

Accepted Answer

The empirical probability of an event is calculated by dividing the number of times the event occurs by the total number of trials. Since the die never came up a 5 (frequency of 5 is 0), the empirical probability of rolling a 5 is 0.

Quiz 7: Probability: Living With the Odds

Use the at least once rule to find the indicated probability. -Find the probability of getting at least one tail when tossing 6 fair coins.

Find the indicated probability. Round your answer to 6 decimal places when necessary. -Three fair coins are tossed. Find the probability of getting the same thing on all three coins.

Solve the problem. -If it has been determined that the probability of an earthquake occurring on a certain day in a certain area is 0.05, what are the odds against an earthquake?

Find the indicated probability. Round your answer to 6 decimal places when necessary. -A class consists of 11 women and 44 men. If a student is randomly selected, what is the probability that the student is a woman?

Find the indicated probability. Round your answer to 6 decimal places when necessary. -One card is selected from a deck of cards. Find the probability of selecting a heart or a card less than 8 . (Note: The ace is considered a low card.)

Determine whether the events A and B are independent. -12 jurors are selected from a pool of 20 Event A: The first person selected is a woman Event B: The second person selected is a woman

Solve the problem. Round your answer to 2 decimal places when necessary. -A U.S. city with a population of 2.2 million reports 30,724 births in one year. How many people are born per day in the city?

Decide whether events A and B are overlapping or non-overlapping. -Event A is that you study at least 5 hours for your math test. Event B is that you study at most 6 hours for your math test.

Solve the problem. -The library is to be given 5 books as a gift. The books will be selected from a list of 21 titles. If each book selected must have a different title, how many possible selections are there?

Solve the problem. Round your answer to 2 decimal places when necessary. -The overall U.S. death rate for 60-62 year-olds is approximately 13 deaths per 1000 people. If there are about 12,142 60-62 year olds in a U.S. city, how many 60-62 year olds can be expected to die in A year in that city?

Find the indicated probability. Round your answer to 6 decimal places when necessary. -A bag contains 6 red marbles, 3 blue marbles, and 7 green marbles. If a marble is randomly selected from the bag, what is the probability that it is blue?

Solve the problem. -License plates are made using 2 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed?

Solve the problem. Round your answer to 2 decimal places when necessary. -The population of a certain country is about 132 million. The overall birth rate is 14.3 births per 1000. Approximately how many births were there in the country?

Find the indicated probability. Round your answer to 6 decimal places when necessary. -Two cards are selected without replacement from a standard deck of 52 cards. What is the probability that both cards are the same color (i.e.either both black or both red) ?

Decide whether events A and B are overlapping or non-overlapping. -A student is selected at random. Event A is that the student speaks Spanish and event B is that the student speaks Chinese.

Find the indicated probability. -A die is rolled 50 times with the following results. $\begin{array} { l | r r r r r r } \text { Outcome } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Frequency } & 7 & 4 & 13 & 23 & 0 & 3\end{array}$ Compute the empirical probability that the die comes up a 5.