Probability of Combined Events (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Probability of Combined Events

What do we mean by combined probabilities?

• In general this means there is more than one event to bear in mind when considering probabilities

  • These events may be independent or mutually exclusive

  • They may involve an event that follows on from a previous event

    • E.g. Rolling a dice, followed by flipping a coin

• Independent events are those whose outcome is not dependent on any other event

  • E.g. "Throwing a heads" on a coin and "rolling a 6" on a dice are independent of each other, one event happening does not affect the other

• Two events are mutually exclusive if they cannot both happen at once

  • For example, when rolling a dice the events “getting a prime number” and “getting a 6” are mutually exclusive

  • Complementary events are mutually exclusive

  • Mutually exclusive events are dependent events

How do I work with and calculate combined probabilities?

• In your head, try to rephrase each question as an AND and/or OR probability statement

  • E.g. The probability of rolling a 6 followed by flipping heads

    • "The probability of rolling a 6 AND the probability of flipping heads"

  • In general,

    • AND means multiply (), this is used for independent events

    • OR mean add (), this is used for mutually exclusive events

• The fact that all probabilities sum to 1 is often used in combined probability questions

  • In particular when we are interested in an event "happening" or "not happening"

    • E.g.  so 

• Tree diagrams can be useful for calculating combined probabilities

  • E.g.  The probability of being stopped at one set of traffic lights and also being stopped at a second set of lights

    • However unless a question specifically tells you to, you don't have to draw a diagram

    • For many questions it is quicker simply to consider the possible options and apply the AND and OR rules without drawing a diagram

How do I calculate combined probabilities for independent events?

• For two events that are independent, you can calculate probabilities for combined events, using the following rules:

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  • On a tree diagram, you would multiply the probabilities along the branches

    • These rules can also be used to test if two events are independent

    • If then and are independent

• Consider the situation of throwing two dice, "getting a 6" on one dice, , "getting an even number" on a second dice,

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• For two independent events, you can find the probability of different possible overall outcomes by adding their individual probabilities:

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  • On a tree diagram, you would add the relevant probabilities down the list of final outcomes

• Consider the previous situation with two dice, and "getting both a 6 and an even" or "not getting a 6 and not getting an even"

  •