Probability of Compound Events

Show your work for each problem.

1.
You flip a coin and roll a die. What is the probability of getting heads AND rolling a 4?
1. a)
2. b)
3. c)
4. d)
Answer:
These are independent events.
2.
Which formula is used for independent events?
1. a)
2. b)
3. c)
4. d)
Answer:
For independent events, we multiply the probabilities: . Addition is used for mutually exclusive OR events.
3.
What is the probability of rolling a 2 OR a 5 on a single die? Give your answer as a simplified fraction (e.g., 1/3).
Answer: 1/3
Rolling a 2 and rolling a 5 are mutually exclusive (can't happen at the same time).
4.
Drawing a card from a deck, putting it back, then drawing another card. Are these events independent or dependent?
1. a)Mutually exclusive
2. b)Dependent
3. c)Independent
4. d)Neither
Answer: Independent
With replacement (putting the card back), the deck is the same for both draws. The first draw does not affect the second, so these are independent events.
5.
You roll two dice. What is the probability of getting a sum of 7? Give your answer as a simplified fraction.
Answer: 1/6
There are 36 total outcomes. The combinations that sum to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 ways.
6.
A bag has 4 red balls and 6 blue balls. You draw two balls without replacement. Find the probability that both balls are blue.
Answer: 1/3
1. How many blue balls out of total for the first draw? 6/10
2. After drawing a blue ball, how many blue are left out of how many total? 5/9
3. Multiply the two probabilities: 6/10 × 5/9 = ? 30/90
4. Simplify 30/90 to lowest terms: 1/3
7.
A spinner has 4 equal sections: red, blue, green, yellow. You spin twice. What is the probability of landing on red both times?
1. a)
2. b)
3. c)
4. d)
Answer:
These are independent events.
8.
A coin is flipped 3 times. What is the probability of getting heads all 3 times? Give your answer as a simplified fraction.
Answer: 1/8
Independent events:
9.
A box contains 3 red pens and 5 blue pens. You randomly pick one pen, do not replace it, then pick another. What is the probability of picking a red pen first AND a blue pen second?
Answer: 15/56
1. What is P(red first)? (red pens / total pens) 3/8
2. After taking a red pen, how many pens are left? What's P(blue second)? 5/7
3. Multiply: 3/8 × 5/7 = ? 15/56
10.
What does P(B|A) represent?
1. a)The probability of B given that A has occurred
2. b)The probability of A and B
3. c)The probability of A given that B has occurred
4. d)The probability of A or B
Answer: The probability of B given that A has occurred
P(B|A) is conditional probability - the probability of event B happening given that event A has already happened. The event after the | is the condition that has occurred.
11.
A jar has 7 green marbles and 3 yellow marbles. You draw 3 marbles without replacement. What is the probability that all 3 are green?
Answer: 7/24
1. What is P(1st green)? 7/10
2. What is P(2nd green | 1st green)? 6/9
3. What is P(3rd green | first two green)? 5/8
4. Multiply all three: 7/10 × 6/9 × 5/8 = ? 210/720
5. Simplify 210/720: 7/24
12.
A password consists of 2 digits (0-9). What is the probability of guessing the password correctly on the first try? Give your answer as a simplified fraction.
Answer: 1/100
There are possible 2-digit passwords. Only 1 is correct, so .
13.
In a class of 30 students, 12 play soccer and 8 play basketball. If the events are independent, what is the probability that a randomly selected student plays both sports?
Answer: 8/75
1. What is P(soccer)? 12/30
2. Simplify 12/30: 2/5
3. What is P(basketball)? 8/30
4. Simplify 8/30: 4/15
5. For independent events, P(both) = 2/5 × 4/15 = ? 8/75
14.
A test is 90% accurate. If you take the test twice (independently), what is the probability that both tests are correct?
1. a)95%
2. b)81%
3. c)180%
4. d)90%
Answer: 81%
. Note that the combined probability is less than each individual test's accuracy.
15.
A bag contains 5 red, 4 green, and 3 blue marbles. You draw 2 marbles without replacement. What is the probability of drawing one red and one green marble (in any order)?
Answer: 10/33
1. P(red first, then green) = P(red) × P(green|red) = ? 5/12 * 4/11
2. Calculate 5/12 × 4/11 = ? 20/132
3. P(green first, then red) = P(green) × P(red|green) = ? 4/12 * 5/11
4. Calculate 4/12 × 5/11 = ? 20/132
5. Add both probabilities: 20/132 + 20/132 = ? 40/132
6. Simplify 40/132: 10/33