GCSE Maths Practice: probability-scale

Understanding Probability Without Replacement

This question focuses on calculating probability when two events happen one after the other and the first event affects the second. In GCSE Maths, this topic is essential because it helps students understand how sample spaces change across sequential events. When cards are drawn without replacement, the total number of available outcomes decreases after each draw. This means each probability must be calculated with updated totals.

A standard deck contains 52 cards divided equally between red cards and black cards. There are 26 red cards (13 hearts and 13 diamonds) and 26 black cards (13 clubs and 13 spades). When the first card is drawn, the chance of selecting a red card is straightforward: there are 26 red cards out of 52. However, when the second card is drawn without replacement, only 51 cards remain in the deck. At this point, if the first card was red, all 26 black cards are still present, so the probability of drawing a black card second is 26 out of 51.

Step-by-Step Method

1. Start by identifying the total number of favourable outcomes for the first event. Here, the favourable outcomes for drawing a red card total 26.
2. Divide by the total number of possible outcomes, which is 52.
3. For the second draw, recognise that one card has been removed. If the first card drawn was red, the blacks remain unchanged at 26, but the total number of cards becomes 51.
4. Express the probability of drawing a black card as \(26/51\).
5. Multiply the two probabilities: \((26/52) \times (26/51)\).

Worked Example 1

Suppose you want the probability of drawing a black card first, followed by a red card. The calculation mirrors the original problem: there are 26 black cards out of 52, then 26 red cards out of 51. Thus, the probability remains \((26/52) \times (26/51)\).

Worked Example 2

If the question asked for drawing two red cards in a row without replacement, the process would change slightly. The first probability remains \(26/52\). However, after removing a red card, only 25 red cards remain out of 51. So the probability becomes \((26/52) \times (25/51)\).

Common Mistakes

• Forgetting to reduce the total number of cards after the first draw.
• Using \(26/52\) again for the second draw instead of updating to \(26/51\).
• Confusing multiplication with addition when combining probabilities. Sequential events require multiplication.

Real-Life Applications

Situations involving selection without replacement appear in quality control, where items are taken from a batch for inspection; in genetics problems where certain traits are selected from a limited pool of alleles; and in games such as drawing tiles, tickets, or tokens from a bag. Understanding the fundamental logic behind changing sample spaces prepares students for more complex statistical questions later in higher education.

FAQ

Q: Why does the total change from 52 to 51?
A: Because the first card is removed and not returned to the deck.

Q: Do we always multiply sequential probabilities?
A: Yes — when events happen in order and both must occur, multiply them.

Q: Does suit matter?
A: No. You only need the total red and black card counts.

Study Tip

Always rewrite the sample space after each draw. Once you get used to checking how many cards remain, probability questions involving multiple events become much easier.