Probability & Types of Events (DP IB Applications & Interpretation (AI)) : 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
  
   
  

  • 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

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:

  •  

    • 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