Question 1

The probability that a certain make of car will need repairs in the first six months is 0.4. A dealer sells seven such cars. What is the probability that at least one of them will require repairs in the first
Six months?

Accepted Answer

The probability that at least one car will need repairs is 1 minus the probability that none of the cars will need repairs. The probability that a car will not need repairs is 0.6. For seven cars, this is $0.6^7 = 0.02799$. Therefore, the probability that at least one will need repairs is $1 - 0.02799 = 0.9720$.

Question 2

A lot of 1000 components contains 300 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. $\text { Find } P ( B \mid A ) \text {. }$

Accepted Answer

Given that the first component drawn is defective, there are now 299 defective components out of a total of 999 components. Therefore, $P(B|A) = \frac{299}{999} \approx 0.2993$.

Question 3

Every day, Bill buys a lottery ticket. Each ticket has probability 0.25 of winning a prize. After seven days, what is the probability that Bill has won at least one prize?

Accepted Answer

The probability that Bill does not win a prize on any given day is 0.75. Therefore, the probability that he does not win a prize on any of the seven days is $(0.75)^7\approx 0.13$. 
Then, the probability that he wins at least one prize is equal to 1 minus the probability that he wins no prizes: $1 - 0.13 = 0.87$. So the answer is D.

Question 4

In the game of tic-tac-toe, if all moves are performed randomly the probability that the game will end in a draw is 0.127. Suppose 12 random games of tic-tac-toe are played. What is the probability
That at least one of them will end in a draw?

Accepted Answer

The probability that none of the 12 games end in a draw is (1-0.127)^12 = 0.196. Therefore, the probability that at least one game ends in a draw is 1-0.196 = 0.8040.

Question 5

Evaluate the factorial: $6 !$

Accepted Answer

6! means 6 factorial, which is equal to 6 x 5 x 4 x 3 x 2 x 1 = 720. Therefore, the correct answer is choice D.

Question 6

A lot of 1000 components contains 200 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).

Accepted Answer

There are 200 defective components out of 1000 total components, so the probability of drawing a defective component on the first try is 200/1000 or 0.2.

Question 7

The numbers 1 through 7 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 $\text { "1", "2", "3", and "4", }$ in that order?

Accepted Answer

The probability of drawing "1", then "2", then "3", then "4" in that order is calculated by multiplying the probabilities of each draw happening in sequence. The probability of drawing "1" first is 1/7, then "2" is 1/6 (since one slip is already removed), then "3" is 1/5, and finally "4" is 1/4. Multiplying these probabilities gives (1/7) * (1/6) * (1/5) * (1/4) = 1/840, which is approximately 0.00119.

Question 8

Evaluate the combination: ${ } _ { 11 } C _ { 7 }$

Accepted Answer

The combination formula is ${ }_nC_r = \frac{n!}{r!(n-r)!}$. For ${ }_{11}C_{7}$, it's $\frac{11!}{7!(11-7)!} = \frac{11!}{7!4!}$. Calculating this gives 330.

Question 9

A study suggests that 14.7% of all four-digit personal identification numbers, or PIN codes, have a repeating-digits format such as 2525. Assuming this to be true, if the PIN codes of four people are
Selected at random, what is the probability that at least one of them will have repeating digits?

Accepted Answer

The answer of A study suggests that 14.7% of all...

Question 10

A lot of 10 components contains 3 that are defective. Two components are drawn at random and tested. Let A be the event that that the first component drawn is defective, and B be the event that
The second component drawn is defective. a. Find P(A).
b. Find $P ( B \mid A )$
c. Find P(A and B).
d. Are A and B independent?

Accepted Answer

P(A) is the probability of drawing a defective component first, which is 3/10 = 0.3000. P(B|A) is the probability of drawing a defective component second given the first was defective, which is 2/9 = 0.2222. P(A and B) is the probability both are defective, which is (3/10)*(2/9) = 0.0667. Events A and B are not independent because P(A and B) ≠ P(A)*P(B).

Question 11

On a TV game show, a contestant is shown 9 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?

Accepted Answer

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

Question 12

Human genetic material (DNA) is made up of sequences of the molecules adenosine (A), guanine (G), cytosine (C), and thymine (T), which are called bases. A codon is a sequence of three bases.
Replicates are allowed, so AAA, CGC, and so forth are codons. Codons are important because each codon causes a different amino acid to be included in a protein.
a. How many different codons are there?
b. How many different codons are there in which all three bases are different?
c. The bases A and G are called purines, while C and T are called pyrimidines. How many different codons are there in which the third base is a purine and the others are pyrimidines?
d. What is the probability that all three bases are different?
e. What is the probability that the third base is a purine and the others are pyrimidines?

Accepted Answer

The answer of Human genetic material (DNA) is made up...

Question 13

At the campus cafeteria, a diner can purchase a "meal deal" that consists of an entree, a side dish, and a dessert. There are 4 choices for the entree, 6 choices for the side dish, and 6 choices for Dessert. How many different meal deals are possible?

Accepted Answer

The number of different meal deals possible is found by multiplying the number of choices for each component of the meal. Thus, the total number of meal deals is 4 (entrees) * 6 (side dishes) * 6 (desserts) = 144.

Question 14

On a TV game show, a contestant is shown 11 products from a grocery store and is asked to choose the three least-expensive items in the set. The three chosen items need not be in any particular order.
In how many ways can the contestant choose the three items?

Accepted Answer

The number of ways to choose 3 items out of 11 is calculated using the combination formula, which is C(n, k) = n! / [k!(n-k)!]. Here, n=11 and k=3, so the calculation is C(11, 3) = 11! / (3! * (11-3)!) = 165.

Question 15

William and Mary are going bowling. The probability that Mary scores more than 175 is 0.6, and the probability that William scores more than 175 is 0.7. Their scores are independent. 
a. Find the probability that both score more than 175 .
b. Given that William scores more than 175 , the probability that Mary scores higher than William is 0.4. Find the probability that William scores more than 175 and Mary scores higher than William.

Accepted Answer

The probability that both score more than 175 is 0.6 * 0.7 = 0.42. Given that William scores more than 175, the probability that Mary scores higher than William is 0.4, so the probability that William scores more than 175 and Mary scores higher than William is 0.7 * 0.4 = 0.28.

Question 16

A committee consists of 10 women and 7 men. Three members are chosen as officers. What is the probability that all three officers are women?

Accepted Answer

The answer of A committee consists of 10 women and...

Question 17

Evaluate the permutation: ${ } _ { 11 } P _ { 6 }$

Accepted Answer

The permutation formula is $nPr = \frac{n!}{(n-r)!}$. Here, $n = 11$ and $r = 6$, so ${ } _ { 11 } P _ { 6 } = \frac{11!}{(11-6)!} = \frac{11!}{5!}$. Calculating this gives 332,640.

Question 18

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. $\text { Find } P ( B \text { and } A ) \text {. }$

Accepted Answer

First, we can find the probability of drawing a defective component on the first draw, which is simply P(A) = 150/1000 = 0.15.

Since the first component drawn is defective, there are now only 999 components left, and 149 of them are defective. Thus, the probability of drawing a defective component on the second draw given that the first component was defective is P(B|A) = 149/999.

To find the probability of both A and B occurring, we use the formula for the probability of the intersection of two events:

P(A and B) = P(A) * P(B|A)

Plugging in our values, we get:

P(A and B) = (150/1000) * (149/999)

Simplifying, we get:

P(A and B) = 0.0224

So the answer is choice A.

Question 19

A computer password consists of nine characters. Replications are allowed.
a. How many different passwords are possible if each character may be any lowercase letter or
digit?
b. How many different passwords are possible if each character may be any lowercase letter?
c. How many different passwords are possible if each character may be any lowercase letter or digit,
and at least one character must be a digit?
d. A computer is generating passwords at random, and each character is equally likely to be any of
the 26 lowercase letters or 10 digits. Replications are allowed. What is the probability that the
password will contain all letters?
e. A computer system requires the password to contain at least one digit. If nine characters are
generated at random from the 26 lowercase letters and 10 digits, what is the probability that they
will form a valid password?

Accepted Answer

The answer of A computer password consists of nine characters....

Quiz 5: Probability

The probability that a certain make of car will need repairs in the first six months is 0.4. A dealer sells seven such cars. What is the probability that at least one of them will require repairs in the first Six months?

A lot of 1000 components contains 300 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. $\text { Find } P ( B \mid A ) \text {. }$

Every day, Bill buys a lottery ticket. Each ticket has probability 0.25 of winning a prize. After seven days, what is the probability that Bill has won at least one prize?

In the game of tic-tac-toe, if all moves are performed randomly the probability that the game will end in a draw is 0.127. Suppose 12 random games of tic-tac-toe are played. What is the probability That at least one of them will end in a draw?

Evaluate the factorial: $6 !$

A lot of 1000 components contains 200 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) .

The numbers 1 through 7 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 $\text { "1", "2", "3", and "4", }$ in that order?

Evaluate the combination: ${ } _ { 11 } C _ { 7 }$

A study suggests that 14.7% of all four-digit personal identification numbers, or PIN codes, have a repeating-digits format such as 2525. Assuming this to be true, if the PIN codes of four people are Selected at random, what is the probability that at least one of them will have repeating digits?

On a TV game show, a contestant is shown 9 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?

At the campus cafeteria, a diner can purchase a "meal deal" that consists of an entree, a side dish, and a dessert. There are 4 choices for the entree, 6 choices for the side dish, and 6 choices for Dessert. How many different meal deals are possible?

On a TV game show, a contestant is shown 11 products from a grocery store and is asked to choose the three least-expensive items in the set. The three chosen items need not be in any particular order. In how many ways can the contestant choose the three items?

A committee consists of 10 women and 7 men. Three members are chosen as officers. What is the probability that all three officers are women?

Evaluate the permutation: ${ } _ { 11 } P _ { 6 }$

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. $\text { Find } P ( B \text { and } A ) \text {. }$

Quiz 5: Probability

The probability that a certain make of car will need repairs in the first six months is 0.4. A dealer sells seven such cars. What is the probability that at least one of them will require repairs in the first Six months?

Every day, Bill buys a lottery ticket. Each ticket has probability 0.25 of winning a prize. After seven days, what is the probability that Bill has won at least one prize?

In the game of tic-tac-toe, if all moves are performed randomly the probability that the game will end in a draw is 0.127. Suppose 12 random games of tic-tac-toe are played. What is the probability That at least one of them will end in a draw?

Evaluate the factorial: 6!6 !6!

A lot of 1000 components contains 200 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) .

The numbers 1 through 7 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", \text { "1", "2", "3", and "4", } "1", "2", "3", and "4", in that order?

Evaluate the combination: 11C7{ } _ { 11 } C _ { 7 }11​C7​

A study suggests that 14.7% of all four-digit personal identification numbers, or PIN codes, have a repeating-digits format such as 2525. Assuming this to be true, if the PIN codes of four people are Selected at random, what is the probability that at least one of them will have repeating digits?

On a TV game show, a contestant is shown 9 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?

At the campus cafeteria, a diner can purchase a "meal deal" that consists of an entree, a side dish, and a dessert. There are 4 choices for the entree, 6 choices for the side dish, and 6 choices for Dessert. How many different meal deals are possible?

On a TV game show, a contestant is shown 11 products from a grocery store and is asked to choose the three least-expensive items in the set. The three chosen items need not be in any particular order. In how many ways can the contestant choose the three items?

A committee consists of 10 women and 7 men. Three members are chosen as officers. What is the probability that all three officers are women?

Evaluate the permutation: 11P6{ } _ { 11 } P _ { 6 }11​P6​

Evaluate the factorial: $6 !$

The numbers 1 through 7 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 $\text { "1", "2", "3", and "4", }$ in that order?

Evaluate the combination: ${ } _ { 11 } C _ { 7 }$

Evaluate the permutation: ${ } _ { 11 } P _ { 6 }$