Sample Space and Events

Show your work for each problem.

1.
What is the sample space when flipping a single coin?
1. a)
2. b)
3. c)
4. d)
Answer:
A coin can only land on heads (H) or tails (T), so the sample space is . 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 .
3.
Which of the following is the event "rolling an even number" on a die?
1. a)
2. b)
3. c)
4. d)
Answer:
The even numbers on a die are 2, 4, and 6. So the event is . Note that 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: total outcomes.
6.
When flipping two coins, which event represents "getting exactly one head"?
1. a)
2. b)
3. c)
4. d)
Answer:
Exactly one head means outcomes with 1 H and 1 T: HT (first H, second T) and TH (first T, second H). So .
7.
A spinner has 4 equal sections: Red, Blue, Green, Yellow. Find the event of landing on a primary color.
Answer: R, B, Y
1. What is the sample space for this spinner? R, B, G, Y
2. What are the primary colors? (from Red, Blue, Green, Yellow) Red, Blue, Yellow
3. Write the event using the first letters. R, B, Y
4. How many outcomes are in this event? 3
8.
How many outcomes are in the sample space when rolling two dice?
Answer: 36
First die: 6 outcomes. Second die: 6 outcomes. Total: 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)Impossible event
3. c)Certain 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: outcomes. The sample space is .
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. List all pairs that sum to 7. (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
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 an even number
2. b)Rolling a prime number
3. c)Rolling a number less than 4
4. d)Rolling a 7
Answer: Rolling a 7
A standard die has faces 1-6, so rolling a 7 is impossible. The event is . The other events are all possible.
14.
A bag contains 3 red marbles and 2 blue marbles. Two marbles are drawn without replacement. Find the sample space.
Answer: RR, RB, BR, BB
1. What colors can the first marble be? R, B
2. If the first marble is red (R), what can the second be? R, B
3. If the first marble is blue (B), what can the second be? R, B
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: . They are: A1, A2, B1, B2, C1, C2.