Question 1

Solve the problem.
-A singer-songwriter wishes to compose a melody. Each note in the melody must be one of the 20 notes in her vocal range. How many different sequences of 5 notes are possible?

Accepted Answer

The answer of Solve the problem.
-A singer-songwriter wishes to compose...

Question 2

Solve the problem.
-How many different 4-letter radio-station call letters can be made if the first letter must be K or W, repeats are allowed, but the call letters cannot end in an O?

Accepted Answer

There are 2 choices for the first letter, 26 choices for the second letter, 26 choices for the third letter, and 25 choices for the fourth letter (since it cannot end in O). So, the total number of different 4-letter radio-station call letters is 2 * 26 * 26 * 25 = 33,800.

Question 3

Solve the problem.
-Find the odds against correctly guessing the answer to a multiple choice question with 7 possible answers.

Accepted Answer

The odds against correctly guessing are the ratio of the number of unsuccessful outcomes to the number of successful outcomes. With 7 possible answers, there is 1 way to be correct and 6 ways to be incorrect, making the odds against 6:1.

Question 4

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-When a pair of dice is rolled there are 36 different possible outcomes: $1 - 1,1 - 2 , \ldots 6$ - If a pair of dice is rolled 4 times, what is the probability of getting a sum of 5 every time?

Accepted Answer

The probability of getting a sum of 5 on a single roll of two dice is $\frac{4}{36}$ or $\frac{1}{9}$, because there are 4 favorable outcomes (1-4, 2-3, 3-2, 4-1) out of 36 possible outcomes. When rolling the dice 4 times, the probability of getting a sum of 5 every time is $(\frac{1}{9})^4 = \frac{1}{6561}$.

Question 5

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-A sample of 4 different calculators is randomly selected from a group containing 42 that are defective and 22 that have no defects. What is the probability that all four of the calculators selected
Are defective?

Accepted Answer

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

Question 6

Solve the problem.
-How many different five-card hands can be dealt from a deck that has only clubs (13 cards altogether)?

Accepted Answer

The number of different five-card hands that can be dealt from a deck of 13 clubs is calculated using the combination formula, which is C(n, k) = n! / [k!(n - k)!], where n is the total number of items (13 clubs) and k is the number of items to choose (5 cards). Thus, C(13, 5) = 13! / [5!(13 - 5)!] = 1,287.

Question 7

Solve the problem.
-A shirt company has 4 designs each of which can be made with short or long sleeves. There are 6 different colors available. How many different shirts are available from this company?

Accepted Answer

Each of the 4 designs can be made with either short or long sleeves, doubling the options to 8 per design. With 6 colors available for each, the total is 8 * 6 = 48 different shirts.

Question 8

Solve the problem.
-How many different three-number "combinations" are possible on a combination lock having 22 numbers on its dial? Assume that no numbers repeat. (Combination locks are really permutation
Locks.)

Accepted Answer

The total number of different three-number "combinations" (permutations, since order matters) possible on a combination lock with 22 numbers, where no numbers repeat, is calculated using the permutation formula $P(n, r) = \frac{n!}{(n-r)!}$. Here, $n = 22$ (total numbers) and $r = 3$ (numbers to choose). Thus, $P(22, 3) = \frac{22!}{(22-3)!} = \frac{22!}{19!} = 22 \times 21 \times 20 = 9240$.

Question 9

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-On a multiple choice test, each question has 6 possible answers. If you make a random guess on the first question, what is the probability that you are correct?

Accepted Answer

The probability of guessing the correct answer randomly when there are 6 options is 1 out of 6, or $\frac{1}{6}$.

Question 10

Solve the problem.
-A sports shop sells tennis rackets in 2 different weights, 3 types of string, and 3 grip sizes. How many different rackets could they sell?

Accepted Answer

The number of different rackets is calculated by multiplying the number of options for each feature together: 2 weights * 3 types of string * 3 grip sizes = 18 different rackets.

Question 11

Find the expected value.
-Bob and Fred play the following game. Bob rolls a single die. If an even number results, Bob must pay Fred the number of dollars indicated by the number rolled. On the other hand, if an odd
Number is rolled, Fred must pay Bob the number of dollars indicated by the number rolled. Find
Bob's expected winnings.

Accepted Answer

The answer of Find the expected value.
-Bob and Fred play...

Question 12

Find the expected value.
-An insurance policy sells for $750. Based on past data, an average of 1 in 80 policyholders will file a $10,000 claim, an average of 1 in 140 policyholders will file a $20,000 claim, and an average of 1 in
200 policyholders will file a $50,000 claim. What is the expected value to the company per policy
Sold?

Accepted Answer

The expected value is the sum of the products of the probabilities and the payoffs. The probability of a $10,000 claim is 1/80, the probability of a $20,000 claim is 1/140, and the probability of a $50,000 claim is 1/200. The expected value is therefore:(1/80) * $10,000 + (1/140) * $20,000 + (1/200) * $50,000 = $232

Question 13

Decide whether events A and B are overlapping or non-overlapping.
-Event A is that I get an A on my spanish test tomorrow. Event B is that I get a B on my spanish test tomorrow.

Accepted Answer

Events A and B are non-overlapping because they cannot both occur at the same time; you cannot get both an A and a B on the same Spanish test.

Question 14

Solve the problem. Round your answer to 5 decimal places when necessary.
-The table shows the leading causes of death for one country in a single recent year. $$\begin{array} { l | r } 
{ \text { Cause of Death } } & \text { Deaths } \
\hline \text { Heart Disease } & 154,800 \
\text { Cancer } & 109,650 \
\text { AIDS } & 57,405 \
\text { Stroke } & 36,120 \
\text { Pulmonary Disease } & 36,249 \
\text { Accidents } & 28,380 \
\text { Diabetes } & 15,609 \
\text { Pneumonia } & 14,448 \
\text { Kidney Disease } & 7611
\end{array}$$ Assume a population of 64.5 million. What is the empirical probability of death by AIDS during a single year?

Accepted Answer

The empirical probability of death by AIDS is calculated by dividing the number of deaths by AIDS (57,405) by the total population (64,500,000). This gives 0.00089 when rounded to five decimal places.

Question 15

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-The probability that Luis will pass his statistics test is 0.75. Find the probability that he will fail his statistics test.

Accepted Answer

The probability of failing is 1 minus the probability of passing, so 1 - 0.75 = 0.25.

Question 16

Make a probability distribution for the given set of events.
-When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below. $( 1,1 ) \quad ( 1,2 ) \quad ( 1,3 ) \quad ( 1,4 ) \quad ( 1,5 ) \quad ( 1,6 )$
$( 2,1 ) \quad ( 2,2 ) \quad ( 2,3 ) \quad ( 2,4 ) \quad ( 2,5 ) \quad ( 2,6 )$
$( 3,1 ) \quad ( 3,2 ) \quad ( 3,3 ) \quad ( 3,4 ) \quad ( 3,5 ) \quad ( 3,6 )$
$( 4,1 ) \quad ( 4,2 ) \quad ( 4,3 ) \quad ( 4,4 ) \quad ( 4,5 ) \quad ( 4,6 )$
$( 5,1 ) \quad ( 5,2 ) \quad ( 5,3 ) \quad ( 5,4 ) \quad ( 5,5 ) \quad ( 5,6 )$
$( 6,1 ) \quad ( 6,2 ) ( 6,3 ) \quad ( 6,4 ) \quad ( 6,5 ) \quad ( 6,6 )$ Let X denote the smaller of the two numbers. If both dice come up the same number, then X equals that common value. Find the probability distribution of X. Leave your probabilities in fraction
Form.

Accepted Answer

To find the probability distribution of X, we count the outcomes where X takes each possible value. For X=1, there are 11 outcomes (all pairs including 1 and pairs of 1,1), making P(X=1) = 11/36. Similarly, for X=2, without counting outcomes already considered for X=1, there are 9 new outcomes, so P(X=2) = 9/36 = 1/4. This pattern continues, adjusting for the decreasing number of outcomes that result in each successive value of X.

Question 17

Determine whether the events A and B are independent.
-Eight friends are drawing straws. The one who picks the short straw must cook dinner for the others. 
Event A: The first person does not pick the short straw
Event B: The second person picks the short straw

Accepted Answer

The outcome of Event A affects the probability of Event B happening because if the first person does not pick the short straw, the probability of the second person picking the short straw increases since there are fewer straws left to choose from.

Question 18

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-You are dealt one card from a 52-card deck. Find the probability that you are not dealt a 5.

Accepted Answer

There are 4 fives in a 52-card deck, so the probability of being dealt a 5 is $\frac{4}{52} = \frac{1}{13}$. Therefore, the probability of not being dealt a 5 is $1 - \frac{1}{13} = \frac{12}{13}$.

Question 19

Find the expected value.
-A commercial building contractor is trying to decide which of two projects to commit her company to.
Project A will yield a profit of $50,000 with a probability of 0.6, a profit of $86,000 with a Probability of 0.3, and a profit of $10,000 with a probability of 0.1.
Project B will yield a profit of $100,000 with a probability of 0.1, a profit of $69,000 with a Probability of 0.7, and a loss of $20,000 with a probability of 0.2.
Find the expected profit for each project. Based on expected values, which project should the Contractor choose?

Accepted Answer

The answer of Find the expected value.
-A commercial building contractor...

Question 20

Find the expected value.
-In a game, you have a 1/43 probability of winning $82 and a 42/43 probability of losing $6. What is your expected winning?

Accepted Answer

The expected value is calculated as (1/43 * $82) + (42/43 * -$6) = $1.91 - $5.86 = -$3.95.

Quiz 7: Probability: Living With the Odds

Solve the problem. -A singer-songwriter wishes to compose a melody. Each note in the melody must be one of the 20 notes in her vocal range. How many different sequences of 5 notes are possible?

Solve the problem. -How many different 4-letter radio-station call letters can be made if the first letter must be K or W, repeats are allowed, but the call letters cannot end in an O?

Solve the problem. -Find the odds against correctly guessing the answer to a multiple choice question with 7 possible answers.

Find the indicated probability. Round your answer to 6 decimal places when necessary. -When a pair of dice is rolled there are 36 different possible outcomes: $1 - 1,1 - 2 , \ldots 6$ - If a pair of dice is rolled 4 times, what is the probability of getting a sum of 5 every time?

Find the indicated probability. Round your answer to 6 decimal places when necessary. -A sample of 4 different calculators is randomly selected from a group containing 42 that are defective and 22 that have no defects. What is the probability that all four of the calculators selected Are defective?

Solve the problem. -How many different five-card hands can be dealt from a deck that has only clubs (13 cards altogether) ?

Solve the problem. -A shirt company has 4 designs each of which can be made with short or long sleeves. There are 6 different colors available. How many different shirts are available from this company?

Solve the problem. -How many different three-number "combinations" are possible on a combination lock having 22 numbers on its dial? Assume that no numbers repeat. (Combination locks are really permutation Locks.)

Find the indicated probability. Round your answer to 6 decimal places when necessary. -On a multiple choice test, each question has 6 possible answers. If you make a random guess on the first question, what is the probability that you are correct?

Solve the problem. -A sports shop sells tennis rackets in 2 different weights, 3 types of string, and 3 grip sizes. How many different rackets could they sell?

Decide whether events A and B are overlapping or non-overlapping. -Event A is that I get an A on my spanish test tomorrow. Event B is that I get a B on my spanish test tomorrow.

Find the indicated probability. Round your answer to 6 decimal places when necessary. -The probability that Luis will pass his statistics test is 0.75. Find the probability that he will fail his statistics test.

Determine whether the events A and B are independent. -Eight friends are drawing straws. The one who picks the short straw must cook dinner for the others. Event A: The first person does not pick the short straw Event B: The second person picks the short straw

Find the indicated probability. Round your answer to 6 decimal places when necessary. -You are dealt one card from a 52-card deck. Find the probability that you are not dealt a 5.

Find the expected value. -In a game, you have a 1/43 probability of winning $82 and a 42/43 probability of losing $6. What is your expected winning?