Probability of Events (SQA National 5 Applications of Mathematics): Revision Note

Probability of events

What are outcomes and events?

• An outcome is a possible result of a trial or experiment

• A sample space is the set of all possible outcomes of an experiment

  • It can be represented as a list or a table

    • e.g. the outcomes when a six-sided dice is rolled are 1, 2, 3, 4, 5, 6

• An event is a collection of outcomes

  • e.g. the dice lands on a six contains the outcome 6

  • e.g. the dice lands on an even number contains the outcomes 2, 4, 6

• An experiment is said to be fair if there is an equal chance of achieving each outcome

  • If there is not an equal chance, the event is biased

  • For example, a fair coin has an equal chance of landing on heads or tails

How do I calculate probabilities of single events with equally likely outcomes?

• STEP 1
  Identify the total number of possible outcomes

  • e.g. there are 49 outcomes when picking a whole number between 1 and 49

• STEP 2
  Identify the number of outcomes for the event

  • e.g. there 9 outcomes that are a multiple of 5

• STEP 3
  Form a fraction for the probability using  

  • e.g. the probability of a number randomly picked from 1 to 49 being a multiple of 5 is

How do I calculate probabilities when items are taken and not replaced?

• It is possible that some items might have been taken and not replaced

  • e.g. when drawing numbers from a bag, the numbers might not be replaced after they are drawn

• Carefully count how many items are remaining

  • e.g. consider a bag with 49 balls numbered 1 to 49

    • if 5 are drawn and not replaced then there are 44 balls left in the bag

• Carefully count how many of the remaining items satisfy the event

  • e.g. consider the event that the 6th ball drawn is an even number

    • there are 24 even numbers in the bag to begin with

    • if three out of the first five are even then there are now 21 even numbers left in the bag

How do I calculate the probability of an event not occurring?

• You might be asked to find the probability that an event does not occur

  • e.g. the probability that a player does not win a game

• One way to do this is to:

  • find the probability that the event does occur

    • e.g. the player wins the game

  • subtract this from 1

• Otherwise, you can calculate the probability directly

  • count the number of outcomes which do not cause the event to occur

  • write as a fraction over the total number of outcomes