Question 1

Two coins are flipped and the number that landed on heads is recorded. The outcomes in the sample space are all equally likely.

Accepted Answer

When two coins are flipped, the sample space consists of the following outcomes: HH, HT, TH, TT. The outcomes HT and TH are not the same event, making the outcomes where one head appears (HT, TH) twice as likely as the other outcomes (HH, TT), thus not all outcomes are equally likely.

Question 2

Use the following to answer the questions:  
You roll two dice and record the sum shown. The sample space is the set of all possible sums, which are 2, 3, . . . , 11, 12.
-What is the complement of the event "the sum is less than 6 or greater than 7"?

Accepted Answer

The complement of the event "the sum is less than 6 or greater than 7" includes all outcomes not in that event, which are the sums equal to 6 and 7.

Question 3

Use the following to answer the questions:  
You roll two dice and record the sum shown. The sample space is the set of all possible sums, which are 2, 3, . . . , 11, 12.
-The probability of the event "the sum is 5" is:

Accepted Answer

There are 36 possible outcomes (6 possibilities for each of the 6 rolls). The number of outcomes where the sum is 5 is 4 (1+4, 2+3, 3+2, 4+1). Therefore, the probability of rolling a sum of 5 is 4/36, which simplifies to 1/9.

Question 4

If you roll two dice and record their sum, the events "even" and "less than or equal to 6" are:

Accepted Answer

The events "even" and "less than or equal to 6" are not disjoint because you can roll sums like 2, 4, or 6, which are both even and ≤ 6. They are not independent because the probability of getting an even number changes if you know the sum is ≤ 6.

Question 5

If you randomly select a committee of two from a group of two men and two women, what is the probability that the group is all female?

Accepted Answer

The answer of If you randomly select a committee of...

Question 6

To function correctly, each of two parts of a circuit must work. If an event denotes that the circuit functions correctly, what is the complement of the event?

Accepted Answer

The complement of the circuit functioning correctly (both parts working) is that it does not function correctly, which occurs if at least one of the two parts is defective.

Question 7

If a fair die is rolled once, what is the probability of getting a number less than five?

Accepted Answer

A fair die has six sides with numbers 1 to 6. Getting a number less than five means getting 1, 2, 3, or 4, which are four possible outcomes out of six equally likely outcomes. Therefore, the probability is 4/6 or 2/3, which is choice C.

Question 8

Three dice are rolled and the number rolled on each die is recorded. The outcomes in the sample space are all equally likely.

Accepted Answer

In a standard dice, each face has an equal probability of coming up, making all outcomes in the sample space equally likely. Thus, the statement is true.

Question 9

Use the following to answer the questions:  
You roll two dice and record the sum shown. The sample space is the set of all possible sums, which are 2, 3, . . . , 11, 12.
-Are the 11 outcomes equally likely?

Accepted Answer

The outcomes are not equally likely, as there are more ways to get a sum of 7 (which is 6 different ways) than there are to get a sum of 2 or 12 (which only have one way each).

Question 10

Three dice are tossed. The sum of the numbers rolled is recorded. How many outcomes are in the sample space?

Accepted Answer

The answer of Three dice are tossed. The sum of...

Question 11

A fair coin is tossed three times. Find the probability of getting exactly two heads.

Accepted Answer

There are a total of $2^3 = 8$ possible outcomes when flipping a coin three times. To get exactly two heads, we can have HHT, HTH, or THH, where H represents heads and T represents tails. Each of these outcomes has a probability of $(1/2)^3 = 1/8$. Therefore, the probability of getting exactly two heads is $3 \cdot (1/8) = 3/8$, making the answer D.

Question 12

Use the following to answer the questions:  
You roll two dice and record the sum shown. The sample space is the set of all possible sums, which are 2, 3, . . . , 11, 12.
-What other outcome has a probability equal to the probability of "the sum is 2"?

Accepted Answer

The probability of rolling a sum of 2 is the same as the probability of rolling a sum of 12, which is only one possible outcome. Therefore, the best choice is C, since the probability of rolling a sum greater than 11 is also only one possible outcome.

Question 13

You use a random number generator that produces the digits 1 through 9 with equal probability. You draw four digits and make them into a number; for example, 0, 2, 4, 4 would be the number 0244 or just 244. What is the probability of getting a number strictly greater than 1000?

Accepted Answer

Since the random number generator produces digits 1 through 9 (excluding 0 for the first digit to make a number strictly greater than 1000), the first digit can be any from 1 to 9, and the subsequent digits can be any from 1 to 9 as well, making every draw greater than 1000 automatically. Therefore, the probability is 1 (or 100%), which is not listed among the options provided.

Question 14

Two fair dice are rolled and the sum rolled is recorded. Find the probability that the sum is 4.

Accepted Answer

To roll a sum of 4, there are three possible outcomes: 1+3, 2+2, and 3+1. Each roll of a fair die has six equally likely outcomes, so there are 6 x 6 = 36 possible outcomes when two dice are rolled. Since there are three ways to roll a sum of 4, the probability of rolling a sum of 4 is 3/36 = 1/12. Therefore, the answer is B.

Question 15

A computer is programmed to randomly print two letters in a row without repeating a letter. What is the probability that the first combination printed is the word "DO"?

Accepted Answer

The total number of possible combinations is 26 (letters in the alphabet) times 25 (remaining letters after the first is chosen), which equals 650. The combination "DO" is just one of these possibilities, so the probability is 1/650.

Question 16

Three dice are rolled and the sum of the numbers rolled is recorded. The outcomes in the sample space are all equally likely.

Accepted Answer

Not all outcomes in the sample space are equally likely. For example, there is only one way to roll a sum of 3, which is to roll three 1s, while there are multiple ways to roll a sum of 7, such as rolling 1-2-4, 2-2-3, 1-3-3, etc.

Question 17

There are seven blue and six black socks in a drawer. One is pulled out at random. Find the probability that it is black.

Accepted Answer

There are a total of thirteen socks in the drawer, six of which are black. Therefore, the probability of pulling out a black sock is 6/13.

Question 18

We need to create three-digit code numbers that must begin with a 7. How many such codes can be made?

Accepted Answer

There are 10 possible digits (0-9) for each of the three digits in the code number. Since the first digit must be 7, there is only 1 choice for the first digit. For the two remaining digits, there are 10 choices for each. Therefore, the total number of possible code numbers is 1 x 10 x 10 = 100.

Question 19

Two coins are flipped and heads or tails is recorded. The outcomes in the sample space are all equally likely.

Accepted Answer

When two coins are flipped, the possible outcomes in the sample space are {HH, HT, TH, TT}, and each outcome is equally likely to occur. Therefore, A is the correct answer.

Question 20

Two dice are rolled and the sum is recorded. What sis the most likely outcome?

Accepted Answer

When two dice are rolled, the possible outcomes are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. Out of these, 7 can be obtained in the most number of ways, specifically six ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). Therefore, the most likely outcome is 7.

Quiz 8: Probability: the Mathematics of Chance