Basic Probability
A fair six-sided die is rolled once. What is the probability of rolling a 5?
1/6
In probability, what does the word OR represent?
The union of two events (A or B or both).
What does the notation P(A | B) represent?
The probability that event A occurs given that event B has already occurred.
What does it mean for two events to be independent?
The occurrence of one event does not change the probability of the other event.
Which rule is used to find the probability that both event A and event B occur?
The multiplication rule.
A bag contains 4 red marbles and 6 blue marbles. One marble is drawn at random. What is the probability the marble is blue?
3/5
Two events A and B have probabilities P(A) = 3/10 and P(B) = 4/10. If the events are mutually exclusive, what is P(A OR B)?
3/10 + 4/10 = 7/10
In a group of 40 students, 24 are seniors. If a student is selected at random, what is P(Senior | Student)?
24/40 = 3/5
What does it mean for two events to be mutually exclusive?
The events cannot occur at the same time, so P(A AND B) = 0.
Two events A and B are mutually exclusive.
If P(A) = 1/3 and P(B) = 1/6, what is P(A OR B)?
1/3 + 1/6 = 1/2
A card is drawn at random from a standard deck of 52 cards. What is the probability the card is not a heart?
3/4
In a group of 50 students, 28 like math, 20 like science, and 8 like both.
What is the probability a randomly selected student likes math OR science?
(28 + 20 − 8) / 50 = 40/50 = 4/5
In a group of 100 students, 60 have a driver’s license, and 30 have both a driver’s license and a job.
What is P(Job | License)?
30/60 = 1/2
If P(A) = 1/4, P(B) = 1/2, and P(A AND B) = 1/8, are events A and B independent? Show how you know.
Yes.
P(A) × P(B) = (1/4)(1/2) = 1/8, which equals P(A AND B).
Two events A and B are independent.
If P(A) = 2/5 and P(B) = 3/10, what is P(A AND B)?
(2/5)(3/10) = 6/50 = 3/25
A spinner has 8 equal sections numbered 1 through 8. What is the probability of landing on an even number or a number greater than 6?
5/8
Why must P(A AND B) be subtracted when calculating P(A OR B) using the Addition Rule?
Because outcomes that are in both A and B are counted twice unless the intersection is subtracted.
Why is P(A | B) calculated by dividing P(A AND B) by P(B)?
Because conditional probability restricts the sample space to outcomes where B has occurred, so probabilities are measured relative to P(B).
Two events A and B are mutually exclusive with P(A) = 2/5 and P(B) = 1/5.
What is P(A AND B), and are the events independent?
P(A AND B) = 0.
The events are not independent because P(A) × P(B) = (2/5)(1/5) = 2/25 ≠ 0.
A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn without replacement.
Which rule should be used to find the probability that both marbles are red, and why?
The multiplication rule using conditional probability, because the probability of the second draw depends on the outcome of the first draw.
A student claims that an event with probability 1/4 is unlikely and therefore will probably not happen in a single trial. Is this reasoning correct? Explain.
No. A probability of 1/4 means the event has a 25% chance of occurring on any single trial. Probability describes long-run behavior, not what must happen in one attempt.
A student calculates P(A OR B) by adding P(A) and P(B) without subtracting anything.
Under what condition would this calculation be correct? Explain.
This calculation is correct only when events A and B are mutually exclusive, meaning they cannot occur together and P(A AND B) = 0.
Let event A be a student has a job, and event B be a student has a driver’s license.
If P(A | B) = 3/5 and P(B) = 2/3, what is P(A AND B)?
P(A AND B) = P(A | B) × P(B)
= (3/5)(2/3)
= 6/15
= 2/5
Let event A be a student passes the math test and event B be a student passes the science test.
Suppose P(A) = 3/5 and P(B) = 4/5, and the two events are independent.
Find P(A AND B).
Since the events are independent,
P(A AND B) = P(A) × P(B)
= (3/5)(4/5)
= 12/25
A card is drawn from a standard deck.
Let event A be the card is a heart, and event B be the card is a face card.
Given:
P(A) = 13/52
P(B) = 12/52
P(A AND B) = 3/52
Find P(A OR B).
P(A OR B) = P(A) + P(B) − P(A AND B)
= 13/52 + 12/52 − 3/52
= 22/52
= 11/26