Sample Space and Events

Show your work for each problem.

1.
What is the sample space when flipping a single coin?
1. a) $S = \{H, T\}$
2. b) $S = \{1, 2\}$
3. c) $S = \{H, T, H, T\}$
4. d) $S = \{H\}$
Answer: $S = \{H, T\}$
A coin can only land on heads (H) or tails (T), so the sample space is $S = \{H, T\}$ . We don't repeat outcomes in a set.
2.
How many outcomes are in the sample space when rolling a standard 6-sided die?
Answer: 6
A standard die has faces numbered 1, 2, 3, 4, 5, and 6. That's 6 possible outcomes, so $n(S) = 6$ .
3.
Which of the following is the event "rolling an even number" on a die?
1. a) $E = \{2, 4, 6\}$
2. b) $E = \{2\}$
3. c) $E = \{1, 3, 5\}$
4. d) $E = \{1, 2, 3, 4, 5, 6\}$
Answer: $E = \{2, 4, 6\}$
The even numbers on a die are 2, 4, and 6. So the event is $E = \{2, 4, 6\}$ . Note that $\{1, 3, 5\}$ would be odd numbers.
4.
List the sample space for flipping two coins.
Answer: HH, HT, TH, TT
1. What are the possible outcomes for the first coin? H, T
2. If the first coin is H, what can the second coin be? H, T
3. List all outcomes starting with H. HH, HT
4. List all outcomes starting with T. TH, TT
5.
A coin is flipped and a die is rolled. How many outcomes are in the sample space?
Answer: 12
The coin has 2 outcomes and the die has 6 outcomes. By the multiplication principle: $2 \times 6 = 12$ total outcomes.
6.
When flipping two coins, which event represents "getting exactly one head"?
1. a) $E = \{HH\}$
2. b) $E = \{HT, TH\}$
3. c) $E = \{HH, HT, TH\}$
4. d) $E = \{TT\}$
Answer: $E = \{HT, TH\}$
Exactly one head means outcomes with 1 H and 1 T: HT (first H, second T) and TH (first T, second H). So $E = \{HT, TH\}$ .
7.
A spinner has 4 equal sections: red, blue, green and yellow. Find the event "not landing on green". How many outcomes does it contain?
Answer: 3
1. How many outcomes are in the sample space? 4
2. Which outcomes are in the event "not green"? red, blue, yellow
3. How many outcomes are in this event? 3
4. How many outcomes of the sample space are NOT in this event? 1
8.
How many outcomes are in the sample space when rolling two dice?
Answer: 36
First die: 6 outcomes. Second die: 6 outcomes. Total: $6 \times 6 = 36$ outcomes in the sample space.
9.
A card is drawn from a standard 52-card deck. How many outcomes are in the event "drawing a face card"?
Answer: 12
1. What are the face cards in a deck? Jack, Queen, King
2. How many face cards are in each suit? 3
3. How many suits are in a deck? 4
4. Total face cards = face cards per suit times number of suits 12
10.
An event that equals the entire sample space is called a:
1. a)Compound event
2. b)Certain event
3. c)Impossible event
4. d)Simple event
Answer: Certain event
When an event equals the sample space, it contains all possible outcomes, so it will certainly happen. This is called a certain event (probability = 1).
11.
Three coins are flipped. How many outcomes are in the sample space?
Answer: 8
Each coin has 2 outcomes. For 3 coins: $2 \times 2 \times 2 = 2^3 = 8$ outcomes. The sample space is $\{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$ .
12.
When rolling two dice, find the number of outcomes where the sum equals 7.
Answer: 6
1. If the first die shows 1, what must the second die show for sum = 7? 6
2. If the first die shows 2, what must the second die show? 5
3. If the first die shows 6, what must the second die show? 1
4. How many outcomes sum to 7? 6
13.
Which of the following is an impossible event when rolling a standard die?
1. a)Rolling a prime number
2. b)Rolling a 7
3. c)Rolling a number less than 4
4. d)Rolling an even number
Answer: Rolling a 7
A standard die has faces 1-6, so rolling a 7 is impossible. The event is $E = \{\} = \emptyset$ . The other events are all possible.
14.
A bag contains 3 red (R) and 2 blue (B) marbles. Two marbles are drawn without replacement. Find the sample space of the colors.
Answer: RR, RB, BR, BB
1. What colors can the first marble be? R, B
2. If the first marble is red, how many red marbles are still in the bag? 2
3. If the first marble is blue, how many blue marbles are still in the bag? 1
4. List all possible outcomes (color pairs). RR, RB, BR, BB
15.
A password consists of one letter (A, B, or C) followed by one digit (1 or 2). How many possible passwords are there?
Answer: 6
Letters: A, B, C (3 choices). Digits: 1, 2 (2 choices). Total passwords: $3 \times 2 = 6$ . They are: A1, A2, B1, B2, C1, C2.