Independent Events

Show your work for each problem.

1.
Which of the following are independent events?
1. a)Flipping a coin twice
2. b)Rolling one die after eating breakfast
3. c)Choosing a student, then choosing another from remaining students
4. d)Drawing two cards without replacement
Answer: Flipping a coin twice
Flipping a coin twice gives independent events - the first flip doesn't affect the second. Drawing cards without replacement and choosing students without replacement are dependent events because the pool changes.
2.
A coin is flipped twice. What is the probability of getting heads both times? Give your answer as a fraction.
Answer: 1/4
3.
What is the probability of rolling a 6 on a die AND getting heads on a coin?
1. a)
2. b)
3. c)
4. d)
Answer:
4.
Two dice are rolled. What is the probability of getting a 1 on both dice?
Answer: 1/36
1. What is P(1 on first die)? 1/6
2. What is P(1 on second die)? 1/6
3. Are these events independent? yes
4. What is ? 1/36
5.
If you flip a coin 3 times, what is the probability of getting heads all 3 times? Give your answer as a fraction.
Answer: 1/8
6.
You have flipped a coin 5 times and got heads every time. What is the probability of heads on the 6th flip?
1. a)
2. b)
3. c)
4. d)Less than because tails is due
Answer:
Each flip is independent! The probability is still . Thinking tails is 'due' is the gambler's fallacy.
7.
You roll a die and flip a coin. What is the probability of getting an even number AND tails?
Answer: 1/4
1. How many even numbers are on a die? 3
2. What is P(even)? 1/2
3. What is P(tails)? 1/2
4. What is P(even) × P(tails)? 1/4
8.
A spinner has 4 equal sections (red, blue, green, yellow). If you spin it twice, what is the probability of getting red both times? Give your answer as a fraction.
Answer: 1/16
9.
A machine has 3 independent parts. Each part has a 90% (0.9) chance of working. What is the probability that all 3 parts work?
Answer: 0.729
1. What is P(part 1 works)? 0.9
2. Are the parts independent? yes
3. Write the multiplication: 0.9 × 0.9 × 0.9 0.9 × 0.9 × 0.9
4. Calculate the result 0.729
10.
Two dice are rolled. What is the probability that both show a number greater than 4?
1. a)
2. b)
3. c)
4. d)
Answer:
. So
11.
A coin is flipped 4 times. What is the probability of getting heads exactly 4 times? Give your answer as a fraction.
Answer: 1/16
12.
A basketball player makes free throws with 80% (0.8) accuracy. What is the probability she makes 3 free throws in a row?
Answer: 0.512
1. What is P(making one free throw)? 0.8
2. Are the free throws independent? yes
3. What is 0.8 × 0.8? 0.64
4. What is 0.64 × 0.8? 0.512
13.
A 4-digit PIN uses digits 0-9. If someone guesses randomly, what is the probability of guessing correctly on the first try?
1. a)
2. b)
3. c)
4. d)
Answer:
14.
A die is rolled and a coin is flipped twice. What is the probability of rolling a 5 AND getting heads on both flips?
Answer: 1/24
1. What is P(rolling a 5)? 1/6
2. What is P(heads on first flip)? 1/2
3. What is P(heads on second flip)? 1/2
4. What is ? 1/24
15.
Each bulb in a string of 5 lights has a 95% (0.95) chance of working. What is the probability all 5 bulbs work? Round to 3 decimal places.
Answer: 0.774
16.
You draw a card from a deck, put it back, and draw again. What is the probability of drawing a heart both times?
Answer: 1/16
1. How many hearts are in a deck? 13
2. How many cards in a deck? 52
3. What is P(heart)? 1/4
4. What is P(heart) × P(heart)? 1/16