Question 1

A fast-food restaurant chain has 633 outlets in the United States. The following table categorizes them by city population and location and presents the number of outlets in each category. An outlet is chosen at random from the 633 to test market a new menu.
$\begin{array}{ccccc}
\hline &{\text { Region }} \
\hline \text { Population } & & & & \
\text { of city } & \text { NE } & \text { SE } & \text { SW } & \text { NW } \
\text { Under } 50,000 & 28 & 30 & 31 & 22 \
50,000-500,000 & 58 & 46 & 54 & 42 \
\text { Over 500,000 } & 79 & 125 & 74 & 44 \
\hline
\end{array}$
Given that the outlet is located in a city with a population under 50,000 , what is the probability that it is in the Southwest?
A) 0.27
B) 0.175
C) 0.279
D) 0.049

Accepted Answer

To find the probability that the outlet is in the Southwest given it is located in a city with a population under 50,000, we use the formula for conditional probability. The number of outlets in cities with a population under 50,000 is 28 (NE) + 30 (SE) + 31 (SW) + 22 (NW) = 111. The number of outlets in the Southwest in cities under 50,000 is 31. Therefore, the probability is 31/111 = 0.279.

Question 2

An unfair coin has a probability 0.6 of landing heads. The coin is tossed four times. What is the probability that it lands heads at least once?
A) 0.25
B) 0.9744
C) 0.8704
D) 0.784

Accepted Answer

The easiest way to solve this problem is to calculate the probability of the coin never landing heads (i.e., landing tails every time) and subtracting that from 1. The probability of landing tails on a single toss is 1 - 0.6 = 0.4. For four tosses, the probability of all tails is $0.4^4 = 0.0256$. Therefore, the probability of getting at least one head is $1 - 0.0256 = 0.9744$.

Question 3

Assume a soldier is selected at random from the Army. Determine whether the events A and B are independent, mutually exclusive, or neither.
A: The soldier is a colonel.
B: The soldier is a sergeant.
A) mutually exclusive
B) independent
C) neither

Accepted Answer

Mutually exclusive events cannot happen at the same time. A soldier cannot be both a colonel and a sergeant simultaneously, making these events mutually exclusive.

Question 4

The letters "A", "B", "C", "D", "E", and "F" are written on six slips of paper, and the slips are placed into a hat. If the slips are drawn randomly without replacement, what is
The probability that "E" is drawn first and "C" is drawn second?
A) 0.028
B) 0.033
C) 0.039
D) 0.024

Accepted Answer

The answer of The letters "A", "B", "C", "D", "E",...

Question 5

The Gift Basket Store had the following premade gift baskets containing the following combinations in stock.
$\begin{array} { l l l l } & \text { Cookies } & \text { Mugs } & \text { Candy } \ \text { coffee } & 5 & 14 & 10 \ \text { Tea } & 16 & 13 & 11 \end{array}$
Choose 1 basket at random. Find the probability that it contains tea given that it contain ns mugs. 
A) $\approx 0.929$
B) $\approx 0.188$
C) $\approx 0.481$
D) $\approx 0.519$

Accepted Answer

The total number of baskets containing mugs is 14 (coffee) + 13 (tea) = 27. The number of those that also contain tea is 13. So, the probability is 13/27, which is approximately 0.481.

Question 6

A fair die is rolled four times. What is the probability that it comes up 1 at least once?
A) 0.1667
B) 0.4213
C) 0.5177
D) 0.8333

Accepted Answer

The answer of A fair die is rolled four times....

Question 7

Given eight students, three of which are female, if two students are selected at
random, without replacement, what is the probability that both students are
female?

Accepted Answer

The answer of Given eight students, three of which are...

Question 8

A lot of 1000 components contains 150 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let
B be the event that the second component drawn is defective.
Find P(A).
A) 0.0067
B) 0.0224
C) 0.15
D) 0.1491

Accepted Answer

P(A) is the probability that the first component drawn is defective, which is simply the ratio of defective components to total components, or 150/1000 = 0.15.

Question 9

A fast-food restaurant chain has 617 outlets in the United States. The following table categorizes them by city population and location and presents the number of outlets in each category. An outlet is chosen at random from the 617 to test market a new menu.
$\begin{array}{ccccc}
\hline & {\text { Region }} \
\hline \text { Population } & & & & \
\text { of city } & \text { NE } & \text { SE } & \text { SW } & \text { NW } \
\text { Under } 50,000 & 27 & 25 & 32 & 22 \
50,000-500,000 & 58 & 45 & 54 & 44 \
\text { Over } 500,000 & 72 & 127 & 70 & 41 \
\hline
\end{array}$
Given that the outlet is located in the West (either SW or NW), what is the probability the is in a city with population $50,000 - 500,000$ ?
A) 0.278
B) 0.764
C) 0.373
D) 0.159

Accepted Answer

First, calculate the total number of outlets in the West (SW + NW). SW = 32 + 54 + 70 = 156, NW = 22 + 44 + 41 = 107, so the total in the West = 156 + 107 = 263. The number of outlets in cities with a population of 50,000 - 500,000 in the West = 54 (SW) + 44 (NW) = 98. The probability = 98 / 263 ≈ 0.373.

Question 10

A recent poll found that 30% of those surveyed are worried about aggressive drivers on the road. If three people are selected at random, what is the probability that all three will
Be worried about aggressive drivers on the road?
A) 0.300
B) 0.027
C) 0.900
D) 0.081

Accepted Answer

The probability that all three selected people are worried about aggressive drivers, given that 30% (or 0.30) of those surveyed are worried, is found by multiplying the individual probabilities together: $0.30 \times 0.30 \times 0.30 = 0.027$.

Question 11

If a coin is tossed three times, the probability that the coin lands on heads at least one of
those three times is 7/8.

Accepted Answer

The probability of not getting heads in three tosses is (1/2)^3 = 1/8. Therefore, the probability of getting at least one head is 1 - 1/8 = 7/8.

Question 12

A group of 10 male and 8 female students is talking about going out for pizza. If 50% of the male students actually go and 25% of the female students actually go, find the
Probability that a random student who goes out for pizza is female. 
A) $\frac { 1 } { 18 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 4 } { 5 }$
D) $\frac { 2 } { 7 }$

Accepted Answer

50% of the male students going means 5 males go, and 25% of the female students going means 2 females go. So, out of the 7 students who go (5 males + 2 females), the probability that a randomly selected student is female is $\frac{2}{7}$.

Question 13

A fair die is rolled two times. What is the probability that both rolls are 1?
A) 0.0046
B) 0.028
C) 0.083
D) 0.167

Accepted Answer

The probability of rolling a 1 on a fair die is 1/6. Since the die is rolled twice, and the rolls are independent, the probability of rolling a 1 both times is (1/6) * (1/6) = 1/36, which is approximately 0.028.

Question 14

Urn 1 contains 3 red balls and 4 black balls. Urn 2 contains 4 red balls and 2 black balls. Urn 3 contains 6 red balls and 5 black balls. If an urn is selected at random and a ball is
Drawn, find the probability it will be red. 
A) $\frac { 13 } { 24 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 379 } { 693 }$
D) $\frac { 13 } { 1386 }$

Accepted Answer

The answer of Urn 1 contains 3 red balls and...

Question 15

A fair coin is tossed four times. What is the probability that the sequence of tosses is HHHT?
A) 0.125
B) 0.038
C) 0.25
D) 0.0625

Accepted Answer

The probability of getting any specific sequence of four coin tosses (like HHHT) is (1/2)^4 = 1/16 = 0.0625, since each toss is independent and has a probability of 1/2.

Question 16

In a second grade class containing 12 girls and 11 boys, 2 students are selected at random to give out the math papers. What is the probability that both are girls? 
A) $\frac { 12 } { 23 } \cdot \frac { 1 } { 2 }$
B) $\frac { 12 } { 23 } \cdot \frac { 12 } { 23 }$
C) $\frac { 11 } { 23 } \cdot \frac { 5 } { 11 }$
D) $\frac { 12 } { 23 } \cdot \frac { 6 } { 11 }$

Accepted Answer

The answer of In a second grade class containing 12...

Question 17

A coin is tossed 3 times. Find the probability that all 3 tosses are tails. 
A) $\frac { 1 } { 6 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 8 }$

Accepted Answer

The probability of getting tails on a single toss is $\frac{1}{2}$. For 3 independent tosses, the probability of all being tails is $\left(\frac{1}{2}\right)^3 = \frac{1}{8}$.

Question 18

The following foreign adoptions (in the United States) occurred during these particular ye
$\begin{array} { l l l } & \mathbf { 2 0 0 6 } & \mathbf { 2 0 1 0 } \ \text { China } & 6677 & 3148 \ \text { Ethiopia } & 755 & 2674 \ \text { Russia } & 3983 & 1366 \end{array}$
Choose one adoption at random from this chart. What is the probability that it was from $\mathrm { E }$ given that it was from 2010 . rs. pia 
A) $\approx 0.780$
B)$ \approx 0.190$
C) $\approx 0.144$
D) $\approx 0.372$

Accepted Answer

The answer of The following foreign adoptions (in the United...

Question 19

A lot of 1000 components contains 250 that are defective. Two components are drawn at random and tested. Let $A$ be the event that the first component drawn is defective, and let $B$ be the event that the second component drawn is defective.
Find $P ( B \mid A )$.
A) 0.2492
B) 0.25
C) 0.0623
D) 0.004

Accepted Answer

Given that the first component drawn is defective, there are now 249 defective components left out of a total of 999 components. Thus, $P(B|A) = \frac{249}{999} = 0.2492$.

Question 20

According to popular belief, 80% of adults enjoy drinking beer. Choose a group of 4 adults at random. The probability that all of them enjoy drinking beer is:
A) 3.200
B) 0.410
C) 0.200
D) 0.250

Accepted Answer

The probability that all 4 adults enjoy drinking beer, given that each has an 80% (or 0.8) chance independently, is calculated by multiplying the probabilities together: $0.8 \times 0.8 \times 0.8 \times 0.8 = 0.4096$, which rounds to 0.410.

Quiz 4: Probability and Counting Rules

An unfair coin has a probability 0.6 of landing heads. The coin is tossed four times. What is the probability that it lands heads at least once?

Assume a soldier is selected at random from the Army. Determine whether the events A and B are independent, mutually exclusive, or neither. A: The soldier is a colonel. B: The soldier is a sergeant.

The letters "A", "B", "C", "D", "E", and "F" are written on six slips of paper, and the slips are placed into a hat. If the slips are drawn randomly without replacement, what is The probability that "E" is drawn first and "C" is drawn second?

A fair die is rolled four times. What is the probability that it comes up 1 at least once?

Given eight students, three of which are female, if two students are selected at random, without replacement, what is the probability that both students are female?

A lot of 1000 components contains 150 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second component drawn is defective. Find P(A) .

A recent poll found that 30% of those surveyed are worried about aggressive drivers on the road. If three people are selected at random, what is the probability that all three will Be worried about aggressive drivers on the road?

If a coin is tossed three times, the probability that the coin lands on heads at least one of those three times is 7/8.

A group of 10 male and 8 female students is talking about going out for pizza. If 50% of the male students actually go and 25% of the female students actually go, find the Probability that a random student who goes out for pizza is female.

A fair die is rolled two times. What is the probability that both rolls are 1?

Urn 1 contains 3 red balls and 4 black balls. Urn 2 contains 4 red balls and 2 black balls. Urn 3 contains 6 red balls and 5 black balls. If an urn is selected at random and a ball is Drawn, find the probability it will be red.

A fair coin is tossed four times. What is the probability that the sequence of tosses is HHHT?

In a second grade class containing 12 girls and 11 boys, 2 students are selected at random to give out the math papers. What is the probability that both are girls?

A coin is tossed 3 times. Find the probability that all 3 tosses are tails.

A lot of 1000 components contains 250 that are defective. Two components are drawn at random and tested. Let $A$ be the event that the first component drawn is defective, and let $B$ be the event that the second component drawn is defective. Find $P ( B \mid A )$ .

According to popular belief, 80% of adults enjoy drinking beer. Choose a group of 4 adults at random. The probability that all of them enjoy drinking beer is: