Probability & Types of Events (DP IB Maths: AI HL)

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

• • 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'

• • • This is in the formula booklet
    • It is commonly written as 

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

• • • 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) )
• The probability of A and B  occurring can be found using the formula

• • • A rearranged version is given in the formula booklet
    • Basically you multiply the probability of A by the probability of B then happening

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:
  •  
    • This is given in the formula booklet
    • This is a useful formula to test whether two events are statistically independent

How do I find the probability of combined mutually exclusive events?

• If A and B are mutually exclusive events then

• • • This is given in the formula booklet
    • This occurs because 
• For any two events A and B the events and  are mutually exclusive and A is the union of these two events
  • 
    • This works for any two events A and B