Question 1

Compute  .

Accepted Answer

The answer of Compute  .
...

Question 2

How many two-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if each digit may be used repeatedly in each number? ​

Accepted Answer

Since each digit can be used repeatedly, there are 6 choices for the first digit and 6 choices for the second digit, making a total of 6 * 6 = 36 possible two-digit numbers.

Question 3

How many three-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if each digit can be used at most once in each number? ​

Accepted Answer

There are six digits available to form three-digit numbers, so we have 6 choices for the first digit, 5 choices for the second digit (since we already used one digit), and 4 choices for the third digit (since we already used two digits). Therefore, the total number of three-digit numbers that can be formed is:

6 x 5 x 4 = 120

So the answer is (E) 120.

Question 4

How many four-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if the numbers are less than 9,000 and digits are not used repeatedly? (Hint: Begin with the digit where there is a restriction on the choices.) ​

Accepted Answer

The first digit has 5 options (1, 3, 5, 7, 8) since the number must be less than 9,000, excluding 9. For the second digit, there are 5 remaining choices (since one digit has been used already), for the third digit there are 4 choices, and for the last digit, there are 3 choices. Multiplying these together gives 5 * 5 * 4 * 3 = 300.

Question 5

A small town has 8000 adult males and 6500 adult females. A sociologist conducted a survey and found that 40% of the males and 30% of the females drink heavily. An adult is selected at random from the town. What is the probability, rounded to four decimal places, that the person is male M, if it is known that the person drinks heavily DH? ​

Accepted Answer

The answer of A small town has 8000 adult males...

Question 6

A small town has 8500 adult males and 5500 adult females. A sociologist conducted a survey and found that 40% of the males and 30% of the females drink heavily. An adult is selected at random from the town. What is the probability, rounded to four decimal places, that the person drinks heavily DH? ​

Accepted Answer

The probability that a randomly selected adult is male is 8500 / (8500 + 5500) = 0.6071. The probability that a randomly selected adult is female is 5500 / (8500 + 5500) = 0.3929. The probability that a randomly selected male drinks heavily is 0.40. The probability that a randomly selected female drinks heavily is 0.30. The probability that a randomly selected adult drinks heavily is (0.6071 * 0.40) + (0.3929 * 0.30) = 0.3607.

Question 7

A small town has 6500 adult males and 5000 adult females. A sociologist conducted a survey and found that 40% of the males and 30% of the females drink heavily. An adult is selected at random from the town. What is the probability, rounded to four decimal places, that the person is male M? ​

Accepted Answer

The probability of selecting a male is the ratio of the number of adult males to the total number of adults: 
Pr(M) = 6500/(6500+5000) = 0.5652 (rounded to four decimal places). The information about heavy drinking is not relevant for this question.

Question 8

The 2003 Toyota Matrix was first made available with 5 different engines and with 6 levels of trim, plus either front-wheel or all-wheel drive. How many different Toyota Matrix cars models were available for purchase? ​

Accepted Answer

To find the total number of available models, we need to multiply the number of engines by the number of trims and then by the number of drive types. 
5 (engines) x 6 (trims) x 2 (drive types) = 60. Therefore, there were 60 different Toyota Matrix car models available for purchase.

Question 9

A small town has 5000 adult males and 7000 adult females. A sociologist conducted a survey and found that 40% of the males and 30% of the females drink heavily. An adult is selected at random from the town. What is the probability, rounded to four decimal places, that the person is male M or drinks heavily DH? ​

Accepted Answer

The probability that the person is male is 5000/(5000+7000) = 5/12. 
The probability that the person drinks heavily is (0.4)*5000/12000 + (0.3)*7000/12000 = 0.35. 
The probability that the person is male or drinks heavily is Pr(M∪DH) = Pr(M) + Pr(DH) - Pr(M∩DH) 
Pr(M∩DH) = Pr(DH| M) * Pr(M) = (0.4)*5000/12000 = 1/6 
Therefore, Pr(M∪DH) = 5/12 + 0.35 - 1/6 = 0.5917 rounded to four decimal places. So, the best choice is C.

Question 10

A self-administered pregnancy test detects 80% of those who are pregnant but does not detect pregnancy in 20%. It is 92% accurate in indicating women who are not pregnant but indicates 8% of this group as being pregnant. Suppose it is known that 2% of the women in a neighborhood are pregnant. If a woman is chosen at random from those living in this neighborhood, and if the test indicates she is pregnant, what is the probability, rounded to four decimal places, that she really is? ​

Accepted Answer

We are looking for the probability that a woman is actually pregnant given that the test indicates she is pregnant. This is a conditional probability, and we can use Bayes' Theorem to solve for it:

Pr(pregnant | tested pregnant) = Pr(tested pregnant | pregnant) * Pr(pregnant) / Pr(tested pregnant)

where
- Pr(pregnant | tested pregnant) is the probability that the woman is actually pregnant given that the test indicates she is pregnant (this is what we want to find)
- Pr(tested pregnant | pregnant) is the probability that the test indicates pregnant given that the woman is actually pregnant (this is the sensitivity of the test, which is 0.8)
- Pr(pregnant) is the overall probability of a woman being pregnant in the neighborhood (which is given as 0.02)
- Pr(tested pregnant) is the overall probability of the test indicating pregnant, regardless of whether the woman is actually pregnant or not

To find Pr(tested pregnant), we need to consider two cases:
- A woman is pregnant and the test indicates she is pregnant (true positive): this happens with probability Pr(tested pregnant | pregnant) * Pr(pregnant) = 0.8 * 0.02 = 0.016
- A woman is not pregnant but the test indicates she is pregnant (false positive): this happens with probability 0.08 * (1 - Pr(pregnant)) = 0.08 * 0.98 = 0.0784

Therefore, Pr(tested pregnant) = 0.016 + 0.0784 = 0.0944

Plugging this and the other given values into Bayes' Theorem, we get:

Pr(pregnant | tested pregnant) = 0.8 * 0.02 / 0.0944 = 0.1695

So the best choice is B.

Question 11

Suppose a box contains 7 defective transistors and 16 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that the first transistor is defective D and the second one is good G? ​

Accepted Answer

The answer of Suppose a box contains 7 defective transistors...

Question 12

The 2003 Toyota Matrix was first made available with 2 different engines and with 4 levels of trim, plus either front-wheel or all-wheel drive. If we include color and there were 8 colors available, how many different Toyota Matrix cars were available for purchase? ​

Accepted Answer

The total number of different Toyota Matrix cars can be calculated by multiplying the number of options for each feature. There are 2 engines, 4 trim levels, 2 types of drive (front-wheel or all-wheel), and 8 colors. So, the total number of combinations is 2 * 4 * 2 * 8 = 128.

Question 13

How many four-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if the numbers mast be even and digits are not repeated? (Hint: Begin with the digit where there is a restriction on the choices.) ​

Accepted Answer

For a number to be even, it must end in an even digit. From the given digits, only 8 is even. So, the last digit is fixed as 8. For the first digit, we have 5 choices (1, 3, 5, 7, or 9), since we can't use 8 again. For the second digit, we have 4 remaining choices (since we've already used one digit for the first position and 8 is fixed for the last), and for the third digit, we have 3 choices. Therefore, the total number of combinations is 5 * 4 * 3 * 1 = 60.

Question 14

If  , find n. ​

Accepted Answer

The code given is a hexadecimal number. We need to convert it to decimal and then find the square root of the result.

11eb0d3b_982d_9e64_9790_05a3d468989b
= 79782457427348557247643840744645171147 (in decimal)

Now, we need to find the square root of this number.

sqrt(79782457427348557247643840744645171147) ≈ 2.81958222103 × 10^23

The decimal part of this number is n. Therefore, n ≈ 2.81958222103 × 10^23.

Rounded to the nearest whole number, n ≈ 22. Thus, the answer is B (22).

Question 15

Compute  . ​

Accepted Answer

The answer of Compute  . ​
...

Question 16

Compute  . ​

Accepted Answer

The answer of Compute  . ​
...

Question 17

Find  . ​

Accepted Answer

The answer of Find  . ​
...

Question 18

Compute  . ​

Accepted Answer

The answer of Compute  . ​
...

Question 19

A survey questioned 1000 people regarding raising the legal drinking age from 18 to 21. Of the 560 people who favored raising the legal drinking age, 390 were female. Of the 440 opposition responses, 120 were female. If a person selected at random from this group is a man, what is the probability that the person favors raising the drinking age? ​

Accepted Answer

The answer of A survey questioned 1000 people regarding raising...

Question 20

Suppose a box contains 6 defective transistors and 13 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that one of the transistors drawn is good G and one of them is defective D? ​

Accepted Answer

The answer of Suppose a box contains 6 defective transistors...

Quiz 8: Introduction to Probability

Compute .

How many two-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if each digit may be used repeatedly in each number? ​

How many three-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if each digit can be used at most once in each number? ​

How many four-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if the numbers are less than 9,000 and digits are not used repeatedly? (Hint: Begin with the digit where there is a restriction on the choices.) ​

A small town has 6500 adult males and 5000 adult females. A sociologist conducted a survey and found that 40% of the males and 30% of the females drink heavily. An adult is selected at random from the town. What is the probability, rounded to four decimal places, that the person is male M? ​

The 2003 Toyota Matrix was first made available with 5 different engines and with 6 levels of trim, plus either front-wheel or all-wheel drive. How many different Toyota Matrix cars models were available for purchase? ​

Suppose a box contains 7 defective transistors and 16 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that the first transistor is defective D and the second one is good G? ​

The 2003 Toyota Matrix was first made available with 2 different engines and with 4 levels of trim, plus either front-wheel or all-wheel drive. If we include color and there were 8 colors available, how many different Toyota Matrix cars were available for purchase? ​

How many four-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if the numbers mast be even and digits are not repeated? (Hint: Begin with the digit where there is a restriction on the choices.) ​

If , find n. ​

Compute . ​

Compute . ​

Find . ​

Compute . ​

Suppose a box contains 6 defective transistors and 13 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that one of the transistors drawn is good G and one of them is defective D? ​

How many two-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if each digit may be used repeatedly in each number?

How many three-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if each digit can be used at most once in each number?

How many four-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if the numbers are less than 9,000 and digits are not used repeatedly? (Hint: Begin with the digit where there is a restriction on the choices.)

A small town has 6500 adult males and 5000 adult females. A sociologist conducted a survey and found that 40% of the males and 30% of the females drink heavily. An adult is selected at random from the town. What is the probability, rounded to four decimal places, that the person is male M?

The 2003 Toyota Matrix was first made available with 5 different engines and with 6 levels of trim, plus either front-wheel or all-wheel drive. How many different Toyota Matrix cars models were available for purchase?

Suppose a box contains 7 defective transistors and 16 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that the first transistor is defective D and the second one is good G?

The 2003 Toyota Matrix was first made available with 2 different engines and with 4 levels of trim, plus either front-wheel or all-wheel drive. If we include color and there were 8 colors available, how many different Toyota Matrix cars were available for purchase?

How many four-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if the numbers mast be even and digits are not repeated? (Hint: Begin with the digit where there is a restriction on the choices.)

If , find n.

Compute .

Compute .

Find .

Compute .

Suppose a box contains 6 defective transistors and 13 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that one of the transistors drawn is good G and one of them is defective D?