Introduction to probability

Section: Probability | Syllabus: Cambridge IGCSE Mathematics (0580)

The Probability Scale and Basic Probability

Probability is measured on a scale from 0 (impossible) to 1 (certain); a probability of 0.5 means the event is just as likely to happen as not (an "even chance")
Probability can be written as a fraction, a decimal, or a percentage - all three are equally acceptable unless a question specifies otherwise
For equally likely outcomes: P(event)=(number of favourable outcomes/total number of possible outcomes)
An outcome must genuinely be equally likely to use this formula directly - if outcomes are not equally likely, this formula cannot be applied without adjustment

Question: Priya writes each of these numbers on a card: -8,\;-3,\;0,\;4,\;7,\;11,\;16,\;29. She picks a card at random. Find the probability that the number on the card is prime.
- Step 1: Identify the prime numbers in the list: 7,\,11,\,29 are prime (3 numbers)
- Step 2: There are 8 numbers in total, all equally likely to be picked
- Answer: P(prime)=(3/8)

MistakeCounting the total number of possible values (e.g. all integers) instead of the total number of actual outcomes given in the question

Fixthe denominator is always the number of outcomes actually available in this situation, not every value that type of number could take

MistakeTreating 0 as prime or forgetting that 1 is not a prime number when counting favourable outcomes

Fixa prime number has exactly two factors (1 and itself); 1 has only one factor and 0 has infinitely many, so neither is prime

The complement of an event is "the event not happening"; since an event either happens or does not, their probabilities always sum to 1: P(not A)=1-P(A)
Events are mutually exclusive if they cannot both happen at the same time (e.g. picking a red counter and picking a blue counter from the same single pick)
If a set of mutually exclusive events covers every possible outcome, their probabilities sum to exactly 1

Question: A bag contains only green, orange and purple counters. Zanele takes a counter at random. P(green)=0.35 and P(orange)=(2/5). Find P(purple).
- Step 1: The three colours are mutually exclusive and cover every possibility, so their probabilities sum to 1: P(green)+P(orange)+P(purple)=1
- Step 2: Convert (2/5) to a decimal to combine with 0.35: (2/5)=0.4
- Step 3: P(purple)=1-0.35-0.4=0.25
- Answer: P(purple)=0.25

MistakeLeaving probabilities in mixed formats (a fraction and a decimal) and trying to subtract them directly

Fixconvert every probability to the same format - usually decimals - before adding or subtracting them

MistakeAssuming the complement rule applies to events that are not actually mutually exclusive and exhaustive

Fixprobabilities only sum to 1 when the listed outcomes cannot overlap and cover every possibility - check this is true before using the rule

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