GCSE Maths Practice: probability-basics

Understanding Complementary Probability

GCSE Maths often includes questions that ask for the probability of an event not happening. This type of reasoning is known as complementary probability. Instead of calculating the probability of an event directly, you find the number of outcomes that are not part of the event. In this question, the event is “drawing a heart,” so the complement is “drawing a card that is not a heart.” Complementary methods are especially useful because they save time and reduce mistakes in more complex probability problems.

Structure of a 52-Card Deck

A standard deck contains four suits: Hearts, Diamonds, Clubs and Spades. Each suit contains exactly 13 cards. Because of this symmetry, once you know the number of cards in one suit, you can easily determine the number of cards belonging to the other three suits.

Counting Non-Hearts

There are 13 hearts in the deck. To find the number of non-hearts, subtract the number of hearts from the full deck:

Non-hearts = 52 − 13 = 39

These 39 cards make up the favourable outcomes for the event “not drawing a heart.”

Probability Formula

The standard formula for simple probability is:

Probability = (Number of favourable outcomes) ÷ (Total number of possible outcomes)

With 39 favourable outcomes and 52 total outcomes, you can apply the formula directly.

Worked Example 1: Basic Complement

Event A: drawing a heart. Complement of A: drawing anything that is not a heart. Since 13 cards are hearts, 39 cards are not. Using complementary reasoning can sometimes be quicker than counting all alternative categories individually.

Worked Example 2: Choosing Another Suit

If the question asked for the probability of drawing a card that is not a club, the method remains the same. There are 13 clubs, so non-clubs = 52 − 13 = 39. This demonstrates how complementary probability generalises across all suits.

Worked Example 3: Removing a Card

If one card has been drawn already and it was a heart, the new deck has 51 cards with only 12 hearts remaining. The number of non-hearts becomes 39 again, but out of 51 this time. Probability changes depending on how many cards remain, which is important in multi-step GCSE problems.

Common Mistakes

• Counting all four suits instead of subtracting one suit from the deck.
• Assuming jokers are included — GCSE questions always use 52 cards.
• Misunderstanding complementary events and mixing them with mutually exclusive events.

Real-Life Applications

Complementary probability is widely used in real situations such as risk analysis, reliability studies and prediction models. For example, if the probability of a machine failing is known, the probability of it working is simply the complement. Understanding this concept early helps with later GCSE and A-level probability topics involving multiple events and tree diagrams.

FAQ

Q: Do I always need to subtract from 52?
No. Subtract from the total number of cards remaining. If cards are removed, adjust the total.

Q: Should I simplify 39/52?
You may if asked, but it is not required unless the question specifies a simplified form.

Q: Why use complementary probability?
It is often faster, especially when calculating the probability of everything except one event.

Study Tip

Whenever you see “not” in a probability question, consider using the complement. Subtract the unwanted outcomes from the full set, then apply the probability formula. This method reduces errors and speeds up calculations in GCSE Maths exams.