Question 1

A bookcase contains 2 statistics books and 5 biology books. If 2 books are chosen at random, the chance that both are statistics books is 
A) $\frac { 10 } { 11 }$
B) $\frac { 1 } { 21 }$
C) $\frac { 10 } { 21 }$
D) $\frac { 1 } { 11 }$

Accepted Answer

The probability of choosing both statistics books is calculated by dividing the number of ways to choose 2 statistics books by the total number of ways to choose any 2 books from the 7 available. The number of ways to choose 2 statistics books is $\binom{2}{2} = 1$, and the total number of ways to choose any 2 books from 7 is $\binom{7}{2} = 21$. Therefore, the probability is $\frac{1}{21}$.

Question 2

A committee consist of 8 women and 11 men. Three members are chosen as officers. What is the probability that all three officers are women?
A) 0.0578
B) 0.01243
C) 0.1703
D) 0.0746

Accepted Answer

The probability that all three officers are women is calculated by dividing the number of ways to choose 3 women from 8 by the total number of ways to choose 3 people from 19 (8 women + 11 men). This is $\frac{{8C3}}{{19C3}} = \frac{{56}}{{969}} \approx 0.0578$.

Question 3

Find the probability of selecting 3 science books and 4 math books from 8 science books and 9 math books. The books are selected at random. 
A) $\approx 0.0065$
B) $\approx 0.0094$
C) $\approx 0.0029$
D) $\approx 0.3628$

Accepted Answer

The probability of selecting 3 science books from 8 science books is $\frac{8!}{3!5!}\times\frac{1}{8^3}$The probability of selecting 4 math books from 9 math books is $\frac{9!}{4!5!}\times\frac{1}{9^4}$The probability of selecting 3 science books and 4 math books is $\frac{8!}{3!5!}\times\frac{1}{8^3}\times\frac{9!}{4!5!}\times\frac{1}{9^4}$$\approx 0.3628$

Question 4

If 20 tickets are sold and 2 prizes are to be awarded, find the probability that one person will win both prizes if that person buys exactly 2 tickets. 
A) $\frac { 1 } { 1140 }$
B) $\frac { 1 } { 480 }$
C) $\frac { 1 } { 190 }$
D) $\frac { 1 } { 380 }$

Accepted Answer

The answer of If 20 tickets are sold and 2...

Question 5

A student and a professor each choose a number between 1 and 9 (1 and 9 are both possible choices). What is the probability that the two choose the same number? 
A) $\frac { 1 } { 81 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 2 } { 81 }$
D) $\frac { 2 } { 9 }$

Accepted Answer

The probability that both the student and the professor choose the same number from 1 to 9 is $\frac{1}{9}$ because there are 9 options and only 1 match for each choice, making the total probability of a match $\frac{1}{9}$.

Question 6

A football team consists of 15 freshmen and 15 sophomores, 14 juniors, and 14 seniors. Four players are selected at random to serve as captains. Find the probability that at least
1 of the students is a senior. 
A) $\approx 0.3200$
B) $\approx 0.9999$
C) $\approx 0.0241$
D) $\approx 0.6800$

Accepted Answer

The probability of selecting at least one senior is 1 minus the probability of selecting no seniors. Calculate the probability of selecting 4 non-seniors from the 44 non-seniors and subtract from 1.

Question 7

A football team consists of 20 freshmen and 20 sophomores, 12 juniors, and 5 seniors. Four players are selected at random to serve as captains. Find the probability that there
Are 2 sophomores and 2 freshmen. 
A) $\approx 0.0914$
B) $ \approx 0.0005$
C) $\approx 0.2000$
D) $ \approx 0.3950$

Accepted Answer

The total number of ways to select 4 players out of 57 is $C(57,4)$. The number of ways to select 2 sophomores out of 20 and 2 freshmen out of 20 is $C(20,2) \times C(20,2)$. The probability is therefore $\frac{C(20,2) \times C(20,2)}{C(57,4)} \approx 0.0914$.

Question 8

In a company there are 8 executives: 5 women and 3 men. 3 are selected to attend a management seminar. Find the probability that 2 men and 1 woman will be selected. 
A) $\approx 0.0667$
B) $0.2679$
C) $\approx 0.3750$
D) $\approx 0.2344$

Accepted Answer

The answer of In a company there are 8 executives:...

Question 9

A package contains 10 resistors, 2 of which are defective. If 3 are selected, find the probability of getting 1 defective resistor. 
A) $\approx 0.3571$
B) $\approx 0.4667$
C) $\approx 0.0167$
D) $\approx 0.2333$

Accepted Answer

The probability of selecting 1 defective resistor out of 3 chosen from 10, where 2 are defective, can be calculated using combinations. The number of ways to choose 1 defective out of 2 is $C(2,1)$, and the number of ways to choose 2 non-defective out of 8 is $C(8,2)$. The total number of ways to choose any 3 resistors out of 10 is $C(10,3)$. Thus, the probability is $\frac{C(2,1) \times C(8,2)}{C(10,3)} = \frac{2 \times 28}{120} = \frac{56}{120} = 0.4667$.

Question 10

The numbers 1 through 9 are written in separate slips of paper, and the slips are placed into a box. Then, 4 of these slips are drawn at random.
What is the probability that the drawn slips are "1", "2", "3", and "4", in that order?
A) 0.007944
B) 0.000331
C) 0.00794
D) 0.19056

Accepted Answer

The probability of drawing "1", then "2", then "3", then "4" in that order is calculated by multiplying the probabilities of drawing each number in sequence. For the first draw, the probability of drawing "1" is 1/9. Assuming "1" is drawn, the probability of then drawing "2" is 1/8, since one slip is already removed. Similarly, the probabilities for "3" and "4" are 1/7 and 1/6, respectively, after the preceding numbers have been drawn. Multiplying these probabilities gives (1/9) * (1/8) * (1/7) * (1/6) = 1/3024, which is approximately 0.000331.

Question 11

When 3 dice are rolled, find the probability of getting a sum of 13. 
A) $\frac { 5 } { 8 }$
B) $\frac { 5 } { 54 }$
C) $\frac { 21 } { 216 }$
D) $\frac { 33 } { 36 }$

Accepted Answer

The total number of outcomes when rolling 3 dice is $6^3 = 216$. To get a sum of 13, we can have combinations like (1,6,6), (2,5,6), (2,6,5), (3,4,6), etc. There are 21 such combinations. Therefore, the probability is $\frac{21}{216}$.

Question 12

Three statistics professors and seven chemistry professors are available to be advisors to a student organization. The student organization needs two of the professors to be
Advisors. If each professor has an equal chance of being selected, what is the probability
That both professors are chemistry professors?
A) 0.467
B) 0.100
C) 0.111
D) 0.233

Accepted Answer

There are 10 professors total, and 7 of them are chemistry professors. The probability of selecting a chemistry professor for the first advisor is 7/10. Once a chemistry professor has been selected, there are 6 chemistry professors remaining out of 9 total professors. The probability of selecting a chemistry professor for the second advisor is 6/9. The probability of selecting two chemistry professors is (7/10) * (6/9) = 0.467.

Question 13

On a TV game show, a contestant is shown 8 products from a grocery store and is asked to choose the three least-expensive items in the set, and then correctly arrange these three
Items in order of price. In how many ways can the contestant choose the three items?
A) 6
B) 6720
C) 56
D) 336

Accepted Answer

The answer of On a TV game show, a contestant...

Quiz 4: Probability and Counting Rules

A bookcase contains 2 statistics books and 5 biology books. If 2 books are chosen at random, the chance that both are statistics books is

A committee consist of 8 women and 11 men. Three members are chosen as officers. What is the probability that all three officers are women?

Find the probability of selecting 3 science books and 4 math books from 8 science books and 9 math books. The books are selected at random.

If 20 tickets are sold and 2 prizes are to be awarded, find the probability that one person will win both prizes if that person buys exactly 2 tickets.

A student and a professor each choose a number between 1 and 9 (1 and 9 are both possible choices) . What is the probability that the two choose the same number?

A football team consists of 15 freshmen and 15 sophomores, 14 juniors, and 14 seniors. Four players are selected at random to serve as captains. Find the probability that at least 1 of the students is a senior.

A football team consists of 20 freshmen and 20 sophomores, 12 juniors, and 5 seniors. Four players are selected at random to serve as captains. Find the probability that there Are 2 sophomores and 2 freshmen.

In a company there are 8 executives: 5 women and 3 men. 3 are selected to attend a management seminar. Find the probability that 2 men and 1 woman will be selected.

A package contains 10 resistors, 2 of which are defective. If 3 are selected, find the probability of getting 1 defective resistor.

The numbers 1 through 9 are written in separate slips of paper, and the slips are placed into a box. Then, 4 of these slips are drawn at random. What is the probability that the drawn slips are "1", "2", "3", and "4", in that order?

When 3 dice are rolled, find the probability of getting a sum of 13.

On a TV game show, a contestant is shown 8 products from a grocery store and is asked to choose the three least-expensive items in the set, and then correctly arrange these three Items in order of price. In how many ways can the contestant choose the three items?