Basic Probability (CIE IGCSE Maths: Core)

What is probability?

• Probability describes the likelihood of something happening
  • In real-life you might use words such as impossible, unlikely and certain
• In maths we use the probability scale to describe probability
  • This means giving it a number between 0 and 1
    • 0 means impossible
    • Between 0 and 0.5 means unlikely
    • 0.5 means even chance
    • Between 0.5 and 1 means likely
    • 1 means certain
• Probabilities can be given as fractions, decimals or percentages

What key words and terminology are used in probability?

• An experiment is an activity that is repeated to produce a set of results
  • Results can be observed (seen) or recorded
  • Each repeat is called a trial
• An outcome is a possible result of a trial
• An event is an outcome (or a collection of outcomes)
  • For example:
    • a dice lands on a six
    • a dice lands on an even number
  • Events are usually given capital letters
  • n(A) is the number of possible outcomes from event A
    
    • A = a dice lands on an even number (2, 4 or 6)
    • n(A) = 3 
• A sample space is the set of all possible outcomes of an experiment
  • It can be represented as a list or a table
• The probability of event A is written P(A)
• An event 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 basic probabilities?

• If all outcomes are equally likely then the probability for each outcome is the same
  • The probability for each outcome is
    • If there are 50 marbles in a bag then the probability of selecting a specific one is
• The theoretical probability of an event can be calculated by dividing the number of outcomes of that event by the total number of outcomes
  •  
  • This can be calculated without actually doing the experiment 

• • • If there are 50 marbles in a bag and 20 are blue, then the probability of selecting a blue marble is

How do I find missing probabilities?

• The probabilities of all the outcomes add up to 1
  • If you have a table of probabilities with one missing, find it by making them all add up to 1 
• The complement of event A is the event where A does not happen
  • This can be thought of as not A
  • P(event does not happen) = 1 - P(event does happen)
    • For example, if the probability of rain is 0.3, then the probability of not rain is 1 - 0.3 = 0.7

What are mutually exclusive events?

• Two events are mutually exclusive if they can not both happen at once
  • 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 the probability of either A or B happening is P(A) + P(B)
• Complementary events are mutually exclusive

What is a possibility diagram?

• In probability, the sample space means all the possible outcomes
• In simple situations it can be given as a list
  • For flipping a coin, the sample space is: Heads, Tails
    • the letters H, T can be used
  • For rolling a six-sided dice, the sample space is:  1, 2, 3, 4, 5, 6 
• If there are two sets of outcomes, a grid can be used
  • These are called possibility diagrams (or sample space diagrams)
  • For example, roll two six-sided dice and add their scores
  • A list of all the possibilities would be very long
    • You might miss a possibility
    • It would be hard to spot any patterns in the sample space

• Combining more than two sets of outcomes must be done by listing the possibilities
  • For example, flipping three coins
    • The sample space is HHH, HHT, HTH, THH, HTT, THT, TTH, TTT (8 possible outcomes)

How do I use a possibility diagram to calculate probabilities?

• Probabilities can be found by counting the number of possibilities you want, then dividing by the total number of possibilities in the sample space
  • In the sample space 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, there are four prime numbers (2, 3, 5 and 7)
    • The probability of getting a prime number is 
  • Using the possibility diagram above for rolling two dice, the probability of getting an eight is 
    • There are 5 eights in the grid, out of the total 36 numbers
• Be careful, this counting method only works if all possibilities in the sample space are equally likely
  • For a fair six-sided dice: 1, 2, 3, 4, 5, 6 are all equally likely
  • For a fair (unbiased) coin: H, T are equally likely
  • Winning the lottery: Win, Lose are are not equally likely! 
    • You cannot count possibilities here to say the probability of winning the lottery is