GCSE Maths Practice: probability-basics

Understanding Probability Without Replacement

This question requires higher-tier GCSE Maths reasoning because it involves dependent events. When cards are drawn from a deck without replacement, the outcome of the first draw affects the probability of the second. After one card is removed, the deck no longer contains 52 cards, and one black card may already be missing. This change in the sample space makes the probability dynamic rather than fixed.

Black Cards in a Standard Deck

A standard 52-card deck has two black suits: clubs and spades. Each suit contains 13 cards, giving a total of 26 black cards. This initial count is essential for the first draw. However, after the first card is drawn and not replaced, the number of cards remaining changes to 51. If the first card was black, the remaining number of black cards becomes 25. These two steps form the basis for dependent probability calculations.

Why the Events Are Dependent

Two events are dependent when the outcome of one affects the probability of the other. Removing a black card on the first draw reduces both the total number of cards and the number of black cards available. This is different from independent events, such as rolling a die and flipping a coin, where one outcome does not influence the other. Recognising dependence is essential when handling sequential event problems.

Worked Example 1: Two Red Cards Without Replacement

A deck has 26 red cards. The probability of drawing a red card first is 26/52. After removing that card, 25 red cards remain out of 51 cards total. The probability of drawing a second red card is 25/51. Multiplying these probabilities gives the combined probability. This example mirrors the structure of black-card problems and helps reinforce the method.

Worked Example 2: Drawing Two Kings Without Replacement

There are 4 Kings in the deck. First draw: 4/52. After drawing a King, only 3 remain in a 51-card deck. Second draw: 3/51. Multiply for the combined probability. This example shows how probabilities dramatically change when dealing with smaller subsets of cards.

Worked Example 3: One Black Card Followed by One Red Card

First draw: black → 26/52. After drawing black, red cards remain at 26 out of 51 total cards. Second draw: red → 26/51. Multiply for the combined probability. This demonstrates mixing event types within dependent scenarios.

Common Mistakes

• Treating dependent events as independent and forgetting the sample space changes.
• Not reducing the deck size from 52 to 51 after the first draw.
• Forgetting to reduce the number of black cards when the first draw is black.
• Multiplying by 26/52 twice, which would only apply to events with replacement.

Real-Life Applications

Dependent-event reasoning appears in quality control (items removed from batches), medical testing (sampling without replacement), card games, probability modelling and simulations. These real-world situations rely on understanding how earlier events affect later ones. This reinforces why the GCSE curriculum introduces dependent probability: it reflects how real systems behave.

FAQ

Q: Why multiply the probabilities?
Because we are finding the probability of both events happening in sequence.

Q: What would change if the cards were replaced?
Both draws would be independent and the deck would reset to 52 cards each time.

Q: Does the order matter?
Yes. Without replacement, the second event always depends on what happened first.

Study Tip

Whenever you see the phrase “without replacement,” immediately adjust the total number of outcomes for the second event. Then multiply the probabilities to find the combined event. This strategy prevents most errors in dependent-event questions.