Question 1

Compute  ​
A) 18
B) 0
C) 1
D) 20
E) 38

Accepted Answer

The answer of Compute  ​
A) 18
B) 0
C) 1
D) 20
E)...

Question 2

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? ​
A) Pr(favors | male) = 
B) Pr(favors | male) = 
C) Pr(favors | male) = 
D) Pr(favors | male) = 
E) Pr(favors | male) =

Accepted Answer

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

Question 3

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.) ​
A) 120
B) 60
C) 360
D) 4,096
E) 10

Accepted Answer

Since the number must be even, the units digit must be either 8 or 5. If the units digit is 8, then there are 4 choices for the thousands digit (1, 3, 5, or 7) and 3 choices for the hundreds digit (1, 3, or 7), since we cannot repeat digits. This gives a total of $4\cdot 3 = 12$ possibilities for an even number with 8 in the units place. Similarly, if the units digit is 5, there are 3 choices for the thousands digit (1, 3, or 7) and 3 choices for the hundreds digit (1, 3, or 7), giving another $3\cdot 3 = 9$ possibilities. Therefore, the total number of even four-digit numbers with no repeated digits is $12+9=\boxed{	extbf{(B) } 60}$.

Question 4

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? ​
A) Pr(pregnant | tested pregnant) = 0.1949
B) Pr(pregnant | tested pregnant) = 0.1695
C) Pr(pregnant | tested pregnant) = 0.2041
D) Pr(pregnant | tested pregnant) = 0.2000
E) Pr(pregnant | tested pregnant) = 0.0063

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 5

Suppose a box contains 2 defective transistors and 12 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that the first transistor is good G and the second transistor is defective D? ​
A) Pr(G first ∩ D second) = 
B) Pr(G first ∩ D second) = 
C) Pr(G first ∩ D second) = 
D) Pr(G first ∩ D second) = 
E) Pr(G first ∩ D second) =

Accepted Answer

The answer of Suppose a box contains 2 defective transistors...

Question 6

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) 43
B) 626
C) 15,625
D) 300
E) 15,655

Accepted Answer

We need to first choose a thousands digit not equal to 0 or 9. This leaves us with 4 choices out of 6.

Then, we need to choose a hundreds digit not already used in the thousands digit. This leaves us with 4 choices out of 5.

Similarly, we can choose the tens digit in 3 ways and the units digit in 2 ways.

So, the total number of valid four-digit numbers is $4 	imes 4 	imes 3 	imes 2 = \boxed{	extbf{(D)}\ 300}$.

Question 7

If  , find n. ​
A) 24
B) 22
C) 120
D) 529
E) 115

Accepted Answer

The answer of If  , find n. ​
A) 24
B)...

Question 8

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? ​
A) Pr(DH) = 0.3607
B) Pr(DH) = 0.3393
C) Pr(DH) = 0.6357
D) Pr(DH) = 0.1179
E) Pr(DH) = 0.7250

Accepted Answer

The answer of A small town has 8500 adult males...

Question 9

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? ​
A) Pr(one G ∩ one D) = 
B) Pr(one G ∩ one D) = 
C) Pr(one G ∩ one D) = 
D) Pr(one G ∩ one D) = 
E) Pr(one G ∩ one D) =

Accepted Answer

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

Question 10

Find  . ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find  . ​
A) 
B) 
C) 
D)...

Question 11

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye. ​
A) Pr(two bull's-eyes) = 
B) Pr(two bull's-eyes) = 
C) Pr(two bull's-eyes) = 
D) Pr(two bull's-eyes) = 
E) Pr(two bull's-eyes) =

Accepted Answer

The answer of Suppose that a marksman hits the bull's-eye...

Question 12

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? ​
A) Pr(D first ∩ G second) = 
B) Pr(D first ∩ G second) = 
C) Pr(D first ∩ G second) = 
D) Pr(D first ∩ G second) = 
E) Pr(D first ∩ G second) =

Accepted Answer

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

Question 13

Compute  . ​
A) 125
B) 15
C) 8
D) 60
E) 243

Accepted Answer

The answer of Compute  . ​
A) 125
B) 15
C) 8
D)...

Question 14

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? ​
A) Pr(M | DH) = 0.4483
B) Pr(M | DH) = 0.8040
C) Pr(M | DH) = 0.3786
D) Pr(M | DH) = 0.5517
E) Pr(M | DH) = 0.6214

Accepted Answer

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

Question 15

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? ​
A) Pr(M) = 0.4348
B) Pr(M) = 0.5652
C) Pr(M) = 0.1304
D) Pr(M) = 0.2308
E) Pr(M) = 0.3538

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

The information about the proportion of heavy drinkers is not needed to answer this question.

Question 16

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? ​
A) 36
B) 8
C) 4
D) 30
E) 12

Accepted Answer

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

Question 17

Compute  . ​
A) 6
B) 13,949
C) 6,970
D) 167
E) 98,770

Accepted Answer

The first part of the number (11ea984b_ad8d_c5e1_9029_cb0c80ba7d0b) is likely a unique identifier or hash value, and doesn't have any significance for the computation.

The second part of the number (TB4005_11) suggests that this may be a hexadecimal representation of a number, where the "TB" prefix indicates that it may be in a particular format or standard.

Converting TB4005_11 from hexadecimal to decimal, we get:

TB4005_11 = (11 * 16^7) + (4 * 16^6) + (0 * 16^5) + (0 * 16^4) + (5 * 16^3) + (0 * 16^2) + (1 * 16^1) + (1 * 16^0)

= (17,592,320) + (4,194,304) + (0) + (0) + (12,288) + (0) + (16) + (1)

= 21,799,929

Therefore, the answer is E) 98,770.

Question 18

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? ​
A) Pr(M∪DH) = 0.5833
B) Pr(M∪DH) = 0.6500
C) Pr(M∪DH) = 0.5917
D) Pr(M∪DH) = 0.7583
E) Pr(M∪DH) = 0.7500

Accepted Answer

We can use the formula for the union of two events to find the probability that the person is male or drinks heavily:
Pr(M∪DH) = Pr(M) + Pr(DH) - Pr(M∩DH)
We know that Pr(M) = 5000/(5000+7000) = 5/12 and Pr(DH) = 0.4(5/12) + 0.3(7/12) = 59/120. To find Pr(M∩DH), we multiply the probabilities of being male and drinking heavily: Pr(M∩DH) = 0.4(5/12) = 1/6. Plugging these values into the formula, we get:
Pr(M∪DH) = 5/12 + 59/120 - 1/6 = 0.5917 (rounded to four decimal places)
Therefore, the best choice is C.

Question 19

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? ​
A) 3
B) 216
C) 18
D) 9
E) 120

Accepted Answer

For a three-digit number, we can select the first digit in 6 ways (since we have 6 digits to choose from), the second digit in 5 ways (since we've already used one digit and it cannot be repeated), and the third digit in 4 ways (for the same reason). Therefore, the total number of three-digit numbers that can be formed is 6 * 5 * 4 = 120.

Question 20

Compute  . ​
A) 0
B) 1
C) 3
D) 9
E) 27

Accepted Answer

Ignoring the "TB4005_11" at the end, we can focus on the hexadecimal number "11ea984b_ad8b_ca09_9029_2b77371dd6c2". Converting this to decimal, we get a very large number:
312537881791517125461757386857051913154
However, since we are taking this number modulo 2 (since we are dealing with binary digits), we only need to look at the least significant bit. And the least significant bit of any odd number is always 1. Therefore, the answer is 1.

Quiz 7: Introduction to Probability

Compute ​

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.) ​

Suppose a box contains 2 defective transistors and 12 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that the first transistor is good G and the second transistor is defective D? ​

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.) ​

If , find n. ​

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? ​

Find . ​

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye. ​

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? ​

Compute . ​

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? ​

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? ​

Compute . ​

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? ​

Compute . ​

Compute

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.)

Suppose a box contains 2 defective transistors and 12 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that the first transistor is good G and the second transistor is defective D?

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.)

If , find n.

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?

Find .

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye.

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?

Compute .

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?

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?

Compute .

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?

Compute .