Independent Events

Show your work for each problem.

1.
Which pair of events is independent?
1. a)Flipping a coin twice
2. b)Spending all day in the sun and getting a sunburn
3. c)Choosing a student, then choosing another from the remaining students
4. d)Drawing two cards without replacement
Answer: Flipping a coin twice
The first coin flip doesn't affect the second, so these events are independent. Drawing cards without replacement and choosing students from the ones left are dependent because the group changes, and a long day in the sun makes a sunburn more likely.
2.
A coin is flipped twice. What is the probability of getting heads both times? Give your answer as a fraction.
Answer: 1/4
$P(\text{heads and heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
3.
What is the probability of rolling a 6 on a die AND getting heads on a coin?
1. a) $\frac{1}{12}$
2. b) $\frac{1}{6}$
3. c) $\frac{1}{3}$
4. d) $\frac{2}{3}$
Answer: $\frac{1}{12}$
$P(6 \text{ and heads}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$
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 $\frac{1}{6} \times \frac{1}{6}$ ? 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
$P(\text{HHH}) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$
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) $\frac{1}{2}$
2. b) $\frac{1}{64}$
3. c) $\frac{1}{32}$
4. d)Less than $\frac{1}{2}$ because tails is due
Answer: $\frac{1}{2}$
Each flip is independent! The probability is still $\frac{1}{2}$ . 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
$P(\text{red and red}) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$
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. What is 0.9 × 0.9? 0.81
4. What is 0.81 × 0.9? 0.729
10.
Two dice are rolled. What is the probability that both show a number greater than 4?
1. a) $\frac{2}{3}$
2. b) $\frac{1}{9}$
3. c) $\frac{4}{9}$
4. d) $\frac{1}{3}$
Answer: $\frac{1}{9}$
$P(>4) = \frac{2}{6} = \frac{1}{3}$ . So $P(\text{both} > 4) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$
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
$P(\text{HHHH}) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16}$
12.
A basketball player makes free throws with 80% (0.8) accuracy. Assume each shot is independent of the others. 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) $\frac{1}{1000}$
2. b) $\frac{1}{10000}$
3. c) $\frac{4}{10}$
4. d) $\frac{1}{40}$
Answer: $\frac{1}{10000}$
$P = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} = \frac{1}{10000}$
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 $\frac{1}{6} \times \frac{1}{2} \times \frac{1}{2}$ ? 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
$P = (0.95)^5 = 0.95 \times 0.95 \times 0.95 \times 0.95 \times 0.95 \approx 0.774$
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