Probabilities of Single Events (College Board AP® Statistics)
Study Guide
Written by: Dan Finlay
Reviewed by: Lucy Kirkham
Sample space
What is a sample space?
A sample space is the set of all possible outcomes of a random process
The outcomes do not overlap
A sample space is written using set notation
e.g. the sample space for rolling a six-sided dice is {1, 2, 3, 4, 5, 6}
Two-way tables and tree diagrams can help to find all outcomes for a sample space when multiple factors are being considered
e.g. a two-way table can be used to find the sample space when a dice is rolled and a coin is flipped
Calculating probabilities of events
How do I find the probability of an event when the outcomes are equally likely?
If the outcomes are equally likely then you can find the probability of an event using the formula
Worked Example
Two fair coins with faces labeled heads (H) and tails (T) are independently flipped. Find the probability that the two coins land on faces with different labels.
Answer:
Find the sample space
The sample space is
Identify the outcomes that are in the event
Let represent the event that both coins land on faces with different labels
The outcomes are equally likely because the coins are fair and flipped independently
Find the probability by using the formula
The probability that the two coins land on faces with different labels is
How do I find the probability of an event when the outcomes are not equally likely?
If the outcomes are not equally likely then you can find the probability of an event by adding together all the probabilities of the outcomes in event
Worked Example
A biased four-sided dice has faces labeled 1 to 4. The dice is rolled and the face it lands on is recorded. The dice landing on a 1, 2, 3 or 4 has a probability of 0.1, 0.2, 0.3 or 0.4 respectively.
Find the probability that the dice lands on a face labeled with a number less than 4.
Answer:
Identify the outcomes that are in the event
Let represent the event that the dice lands on a face labeled with a number less than 4
The outcomes are not equally likely so add the probabilities of each outcome in the event
The probability that the dice lands on a face labeled with a number less than 4 is 0.6
Complement of an event
What is the complement of an event?
The complement of an event is the set of outcomes that are not in the event
If event is a dice landing on an even number when rolled
then its complement is the event that the dice lands on an odd number
The complement of an event is denoted by or
How do I find the probability of the complement of an event?
The probabilities of an event and its complement add up to 1
Subtract the probability of an event from 1 to find the probability of its complement
Examiner Tips and Tricks
If you see the phrase "at least" in an exam question, then consider using the complement of an event.
Worked Example
Jim plays a game once each day. The probability that Jim first wins the game on day 1 is 0.5. The probability that Jim first wins the game on any given day is half the probability that Jim first wins the game on the previous day. For example, the probability that Jim wins the game on day 2 is 0.25.
Find the probability that Jim does not win his first game until at least day 3.
Answer:
Identify the event
Let represent the event that Jim does not win his first game until at least day 3
The event has an infinite number of outcomes as Jim could win his first game on day 3, day 4, day 5, and so on
The missing outcomes are that Jim first wins on day 1 and that Jim first wins on day 2, this is the complement of the event in the question
is the event that Jim first wins the game on day 1 or day 2
Find the probability of the complement by adding together the probability that Jim first wins on day 1 and the probability that Jim first wins on day 2
Subtract this probability from 1 to find the probability that Jim does not win his first game until at least day 3
The probability that Jim does not win his first game until at least day 3 is 0.25
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