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Probability & Types of Events (DP IB Maths: AA HL)
Revision Note
Probability Basics
What key words and terminology are used with probability?
- An experiment is a repeatable activity that has a result that can be observed or recorded
- Trials are what we call the repeats of the experiment
- An outcome is a possible result of a trial
- An event is an outcome or a collection of outcomes
- Events are usually denoted with capital letters: A, B, etc
- n(A) is the number of outcomes that are included in event A
- An event can have one or more than one outcome
- A sample space is the set of all possible outcomes of an experiment
- This is denoted by U
- n(U) is the total number of outcomes
- It can be represented as a list or a table
How do I calculate basic probabilities?
- If all outcomes are equally likely then probability for each outcome is the same
- Probability for each outcome is
- Theoretical probability of an event can be calculated without using an experiment by dividing the number of outcomes of that event by the total number of outcomes
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- This is given in the formula booklet
- Identifying all possible outcomes either as a list or a table can help
- Experimental probability (also known as relative frequency) of an outcome can be calculated using results from an experiment by dividing its frequency by the number of trials
- Relative frequency of an outcome is
How do I calculate the expected number of occurrences of an outcome?
- Theoretical probability can be used to calculate the expected number of occurrences of an outcome from n trials
- If the probability of an outcome is p and there are n trials then:
- The expected number of occurrences is np
- This does not mean that there will exactly np occurrences
- If the experiment is repeated multiple times then we expect the number of occurrences to average out to be np
What is the complement of an event?
- The probabilities of all the outcomes add up to 1
- Complementary events are when there are two events and exactly one of them will occur
- One event has to occur but both events can not occur at the same time
- The complement of event A is the event where event A does not happen
- This can be thought of as not A
- This is denoted A'
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- This is in the formula booklet
- It is commonly written as
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What are different types of combined events?
- The intersection of two events (A and B) is the event where both A and B occur
- This can be thought of as A and B
- This is denoted as
- The union of two events (A and B) is the event where A or B or both occur
- This can be thought of as A or B
- This is denoted
- The event where A occurs given that event B has occurred is called conditional probability
- This can be thought as A given B
- This is denoted
How do I find the probability of combined events?
- The probability of A or B (or both) occurring can be found using the formula
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- This is given in the formula booklet
- You subtract the probability of A and B both occurring because it has been included twice (once in P(A) and once in P(B) )
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- The probability of A and B occurring can be found using the formula
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- A rearranged version is given in the formula booklet
- Basically you multiply the probability of A by the probability of B then happening
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Examiner Tip
- In an exam drawing a Venn diagram or tree diagram can help even if the question does not ask you to
Worked example
Dave has two fair spinners, A and B. Spinner A has three sides numbered 1, 4, 9 and spinner B has four sides numbered 2, 3, 5, 7. Dave spins both spinners and forms a two-digit number by using the spinner A for the first digit and spinner B for the second digit.
is the event that the two-digit number is a multiple of 3.
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Independent & Mutually Exclusive Events
What are mutually exclusive events?
- Two events are mutually exclusive if they cannot both occur
- For example: when rolling a dice the events "getting a prime number" and "getting a 6" are mutually exclusive
- If A and B are mutually exclusive events then:
What are independent events?
- Two events are independent if one occurring does not affect the probability of the other occurring
- For example: when flipping a coin twice the events “getting a tails on the first flip” and “getting a tails on the second flip” are independent
- If A and B are independent events then:
- and
- If A and B are independent events then:
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- This is given in the formula booklet
- This is a useful formula to test whether two events are statistically independent
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How do I find the probability of combined mutually exclusive events?
- If A and B are mutually exclusive events then
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- This is given in the formula booklet
- This occurs because
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- For any two events A and B the events and are mutually exclusive and A is the union of these two events
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- This works for any two events A and B
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Worked example
Find the probability that the student has both a dog and a cat.
Given that and are independent, find .
Given that and are mutually exclusive and that find and .
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