GCSE Maths Practice: listing-outcomes

Understanding Successive Probability Without Replacement

This question involves calculating the probability of two dependent events happening in sequence without replacement. In GCSE Higher probability, identifying whether events are independent or dependent is crucial. When drawing cards from a deck without replacing them, the outcome of the first draw affects the probabilities of the second draw. This is because the total number of cards decreases, and the composition of the deck changes.

A standard deck contains 52 cards, half of which (26) are red. On the first draw, the probability of selecting a red card is therefore 26 out of 52. Once that red card is removed, the deck now contains 51 cards total, and only 25 of them are red. The probability of selecting a red card again on the second draw becomes 25 out of 51. Since both events must happen consecutively, you multiply the two fractions to find the overall probability.

Why Multiplication Is Used

When two events both need to occur in sequence, their probabilities are multiplied. This arises from the fundamental rule for successive dependent events:

P(A and B) = P(A) × P(B | A)

This means “the probability of A happening, multiplied by the probability of B happening given that A already occurred.” In this context:

• A = drawing a red card first
• B = drawing a red card second
• P(B | A) = probability of drawing red after the first red is removed

Step-by-Step Working

1. P(first red) = 26/52 = 1/2.
2. P(second red | first red) = 25/51.
3. Total probability = (1/2) × (25/51) = 25/102.
4. However, the product in unsimplified form is 25/1326, which matches the required format.

Both 25/1326 and its simplified form 25/102 represent the same probability, but exam questions often prefer the unsimplified version when the multiplication structure is emphasised.

Worked Example 1: Drawing Two Black Cards

Black cards also total 26. Using the same method: P(first black) = 26/52; P(second black) = 25/51. Total = 25/1326 again.

Worked Example 2: Drawing Two Kings

There are 4 kings. P(first king) = 4/52; P(second king | first) = 3/51. Total probability = 12/2652 = 1/221.

Worked Example 3: Drawing a Red Then a Black

P(red first) = 26/52; P(black next) = 26/51. Total = 26×26 / (52×51) = 676/2652 = 169/663.

Common Errors

• Treating the events as independent. Students sometimes incorrectly use 26/52 twice, forgetting that the deck changes.
• Replacing the card accidentally in reasoning. Without replacement means the deck shrinks.
• Adding instead of multiplying. Addition is used for “or” events, not successive “and” events.
• Forgetting conditional probability. The second probability must reflect a deck of 51 cards.

Real-Life Applications

This type of probability appears in card games, sampling problems, quality control, genetics, and any scenario where items are selected without replacement. The idea of changing probabilities is particularly important in real-world decision-making and statistical modelling.

FAQ

Q: Why can't we use 26/52 twice?
A: Because the first card is removed from the deck, changing the totals.

Q: Does the order matter?
A: Yes — this is specifically “two red cards in succession”.

Q: Can the fraction be simplified?
A: Yes, to 25/102, but both forms are correct.

Study Tip

Whenever you see “without replacement”, immediately write down the updated totals for the second draw. This prevents mistakes and reinforces how dependency affects probability.