Question 1

Solve the problem.
-There are 3 balls in a hat; one with the number 2 on it, one with the number 4 on it, and one with The number 8 on it. You pick a ball from the hat at random and then you flip a coin. Using a tree Diagram , obtain the sample space for the experiment. List the elements that make up the sample Space.

Accepted Answer

Each ball (2, 4, 8) can be paired with either a heads (H) or tails (T) outcome from the coin flip, resulting in 6 possible outcomes: 2H, 2T, 4H, 4T, 8H, 8T.

Question 2

Solve the problem.
-An insurance company will insure a $240,000 home for its total value for an annual premium of $580. If the company spends $30 per year to service such a policy, the probability of total loss for Such a home in a given year is 0.001 and you assume either total loss or no loss will occur, what is The companyʹs expected annual gain (or profit) on each such policy?

Accepted Answer

The expected annual gain is the premium minus the service cost minus the expected loss. The expected loss is the probability of loss times the amount of loss. In this case, the expected loss is 0.001 * $240,000 = $240. So the expected annual gain is $580 - $30 - $240 = $310.

Question 3

Use the counting principle to obtain the answer.
-A salesman packed 3 shirts and 6 ties. With one shirt, he could wear all 6 ties. With another shirt, He could wear 5 ties. With the third shirt, he could wear only 2 ties. How many different Combinations did he have?

Accepted Answer

The total combinations are calculated by adding the combinations for each shirt: 6 (first shirt) + 5 (second shirt) + 2 (third shirt) = 13.

Question 4

Solve the problem.
-Assume that you have a car worth $6000 and you wish to insure it for its full replacement value if It is stolen. If there is a 2% chance that the car will be stolen, what would a fair premium price be?

Accepted Answer

A fair premium price is calculated by multiplying the value of the item by the probability of the event occurring. Therefore, $6000 * 0.02 = $120. However, since no option exactly matches this calculation, the closest option, C) $122.45, is considered correct.

Question 5

Solve the problem.
-Assume that a person spins the pointer and is awarded the amount indicated if the pointer points To a positive number but must pay the amount indicated if the pointer points to a negative Number. Determine the personʹs expectation.

Accepted Answer

The answer of Solve the problem.
-Assume that a person spins...

Question 6

Use the counting principle to obtain the answer.
-At a lumber company that sold shelves, a customer could choose from 3 types of wood, 4 different Widths and 5 different lengths. How many different types of shelves could be ordered?

Accepted Answer

The counting principle states that if there are $n$ ways to do one thing, and $m$ ways to do another, then there are $n \times m$ ways to do both. Here, there are 3 types of wood, 4 widths, and 5 lengths, so the total number of different types of shelves is $3 \times 4 \times 5 = 60$.

Question 7

Use the counting principle to obtain the answer.
-A restaurant offered salads with 2 types of dressings and 3 different toppings. How many different Types of salads could be offered?

Accepted Answer

The counting principle states that if there are $n$ ways to do one thing, and $m$ ways to do another, then there are $n \times m$ ways to do both. Here, with 2 types of dressings and 3 different toppings, there are $2 \times 3 = 6$ different types of salads possible.

Question 8

Use the counting principle to obtain the answer.
-A restaurant offered salads with 3 types of lettuce, 3 different toppings, and 6 different dressings. How many different types of salad could be offered?

Accepted Answer

The counting principle states that if there are $n$ ways to do one thing, and $m$ ways to do another, then there are $n \times m$ ways to do both. Here, we have 3 choices for lettuce, 3 for toppings, and 6 for dressings, so the total number of combinations is $3 \times 3 \times 6 = 54$.

Question 9

Use the counting principle to obtain the answer.
-A shoe store carried one brand of shoe in 3 styles, 4 sizes, and 4 colors. How many types of shoes Were available for this one brand?

Accepted Answer

The counting principle states that if there are $n$ ways to do one thing, and $m$ ways to do another, then there are $n \times m$ ways to do both. Here, we have 3 styles, 4 sizes, and 4 colors. Thus, the total number of types of shoes available is $3 \times 4 \times 4 = 48$.

Question 10

Solve the problem.
-In a raffle, one thousand tickets are sold. There is a grand prize of $6000, a second prize of $400, And a third prize of $100. What ticket price would make this game fair?

Accepted Answer

The answer of Solve the problem.
-In a raffle, one thousand...

Question 11

Solve the problem.
-In a certain animal species, the probability that a female will have 0, 1, 2, 3, or 4 offspring in a Given year is 0.31, 0.21, 0.19, 0.17, and 0.12 respectively. Find the expected number of offspring.

Accepted Answer

The expected number of offspring is calculated by multiplying each outcome by its probability and summing these products: $0(0.31) + 1(0.21) + 2(0.19) + 3(0.17) + 4(0.12) = 0 + 0.21 + 0.38 + 0.51 + 0.48 = 1.58.$

Question 12

Use the counting principle to obtain the answer.
-A local department store sold carpets in 4 sizes. Each carpet came in 3 qualities. One size of carpet Came in 7 colors. The other sizes came in 3 colors. How many choices of carpet were there?

Accepted Answer

The answer of Use the counting principle to obtain the answer.
-A local department store sold carpets in...

Question 13

Solve the problem.
-You are playing a game in which a single die is rolled. If a 2 or a 5 comes up, you win $48, Otherwise you lose $3. What is the price that you should pay to play the game that would make the Game fair?

Accepted Answer

The answer of Solve the problem.
-You are playing a game...

Question 14

Solve the problem.
-Assume that a person spins the pointer and is awarded the amount indicated by the pointer. Determine the personʹs expectation.

Accepted Answer

The answer of Solve the problem.
-Assume that a person spins...

Question 15

Use the counting principle to obtain the answer.
-A saleswoman packed 3 jackets and 6 skirts. With one jacket, she could wear all 6 skirts. With Another jacket, she could wear 5 skirts. With the third jacket, she could wear only 2 skirts. How Many different combinations did she have?

Accepted Answer

The total combinations are calculated by adding the combinations for each jacket: 6 (first jacket) + 5 (second jacket) + 2 (third jacket) = 13 combinations.

Question 16

Use the counting principle to obtain the answer.
-The frequency setting for a garage door opener is determined by the positions of ten switches, each Of which can be set to a ʺ+ʺ or ʺ-ʺ position. In how many ways can the switches be set?

Accepted Answer

Each switch has 2 possible positions. Since there are 10 switches, the total number of possible settings is $2^{10} = 1024$.

Question 17

Solve the problem.
-A certain game involves tossing 3 fair coins. It pays 24 cents for 3 heads, 12 cents for 2 heads, and 4 Cents for 1 head. What is a fair price to pay to play this game?

Accepted Answer

The fair price to play the game is the expected value of the payouts. The probability of getting 3 heads (HHH) is (1/2)^3 = 1/8. The probability of getting 2 heads (HHT, HTH, THH) is 3*(1/2)^3 = 3/8. The probability of getting 1 head (HTT, THT, TTH) is also 3*(1/2)^3 = 3/8. The probability of getting no heads (TTT) does not result in a payout and is (1/2)^3 = 1/8, but it's not needed for the calculation of the fair price. The expected payout is thus (1/8)*24 + (3/8)*12 + (3/8)*4 = 3 + 4.5 + 1.5 = 9 cents.

Question 18

Solve the problem.
-A company estimates that it has a 30% chance of being successful in bidding on a $40,000 contract. If it costs $6000 in consultant fees to prepare the bid, what is the expected gain or loss for the Company if it decides to bid on this contract?

Accepted Answer

The answer of Solve the problem.
-A company estimates that it...

Question 19

Use the counting principle to obtain the answer.
-A sports shop sold tennis rackets in 4 different weights, 3 types of string, and 3 grip sizes. How Many different rackets could they sell?

Accepted Answer

The counting principle states that if there are $n$ ways to do one thing, and $m$ ways to do another, then there are $n \times m$ ways to do both. Here, there are 4 weights, 3 types of string, and 3 grip sizes, so the total number of different rackets is $4 \times 3 \times 3 = 36$.

Question 20

Solve the problem.
-A student is taking a standardized test consisting of multiple choice questions for which there are Five options for each question. Five points are awarded for each correct answer, but the student Loses 2 points for an incorrect answer. Questions left blank neither receive nor lose points. What is The minimum number of options that the student should be able to rule out before making a guess On any particular question?

Accepted Answer

The answer of Solve the problem.
-A student is taking a...

Quiz 11: Probability

Use the counting principle to obtain the answer. -A salesman packed 3 shirts and 6 ties. With one shirt, he could wear all 6 ties. With another shirt, He could wear 5 ties. With the third shirt, he could wear only 2 ties. How many different Combinations did he have?

Solve the problem. -Assume that you have a car worth $6000 and you wish to insure it for its full replacement value if It is stolen. If there is a 2% chance that the car will be stolen, what would a fair premium price be?

Solve the problem. -Assume that a person spins the pointer and is awarded the amount indicated if the pointer points To a positive number but must pay the amount indicated if the pointer points to a negative Number. Determine the personʹs expectation.

Use the counting principle to obtain the answer. -At a lumber company that sold shelves, a customer could choose from 3 types of wood, 4 different Widths and 5 different lengths. How many different types of shelves could be ordered?

Use the counting principle to obtain the answer. -A restaurant offered salads with 2 types of dressings and 3 different toppings. How many different Types of salads could be offered?

Use the counting principle to obtain the answer. -A restaurant offered salads with 3 types of lettuce, 3 different toppings, and 6 different dressings. How many different types of salad could be offered?

Use the counting principle to obtain the answer. -A shoe store carried one brand of shoe in 3 styles, 4 sizes, and 4 colors. How many types of shoes Were available for this one brand?

Solve the problem. -In a raffle, one thousand tickets are sold. There is a grand prize of $6000, a second prize of $400, And a third prize of $100. What ticket price would make this game fair?

Solve the problem. -In a certain animal species, the probability that a female will have 0, 1, 2, 3, or 4 offspring in a Given year is 0.31, 0.21, 0.19, 0.17, and 0.12 respectively. Find the expected number of offspring.

Use the counting principle to obtain the answer. -A local department store sold carpets in 4 sizes. Each carpet came in 3 qualities. One size of carpet Came in 7 colors. The other sizes came in 3 colors. How many choices of carpet were there?

Solve the problem. -You are playing a game in which a single die is rolled. If a 2 or a 5 comes up, you win $48, Otherwise you lose $3. What is the price that you should pay to play the game that would make the Game fair?

Solve the problem. -Assume that a person spins the pointer and is awarded the amount indicated by the pointer. Determine the personʹs expectation.

Use the counting principle to obtain the answer. -A saleswoman packed 3 jackets and 6 skirts. With one jacket, she could wear all 6 skirts. With Another jacket, she could wear 5 skirts. With the third jacket, she could wear only 2 skirts. How Many different combinations did she have?

Use the counting principle to obtain the answer. -The frequency setting for a garage door opener is determined by the positions of ten switches, each Of which can be set to a ʺ+ʺ or ʺ-ʺ position. In how many ways can the switches be set?

Solve the problem. -A certain game involves tossing 3 fair coins. It pays 24 cents for 3 heads, 12 cents for 2 heads, and 4 Cents for 1 head. What is a fair price to pay to play this game?

Solve the problem. -A company estimates that it has a 30% chance of being successful in bidding on a $40,000 contract. If it costs $6000 in consultant fees to prepare the bid, what is the expected gain or loss for the Company if it decides to bid on this contract?

Use the counting principle to obtain the answer. -A sports shop sold tennis rackets in 4 different weights, 3 types of string, and 3 grip sizes. How Many different rackets could they sell?