A-Level — mathsStandard1 min read

Probability and Conditional Probability

Apply probability rules at A-Level including conditional probability, independence, and tree diagrams.

Tutor AI Team2026-02-25T01:58:08.630Z

A-Level probability extends GCSE with formal notation, conditional probability, and independence tests.

Core Formulas

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

$P(A \cap B) = P(A|B) \times P(B) = P(B|A) \times P(A)$

A and B are independent if $P(A \cap B) = P(A) \times P(B)$ , equivalently $P(A|B) = P(A)$ .

$P(A \cap B) = 0$ . Cannot both happen.

Branch probabilities are conditional. Multiply along, add between paths.

$P(A) = 0.4$ , $P(B) = 0.5$ , $P(A \cap B) = 0.2$ .

$P(A \cup B) = 0.7$ . $P(A|B) = \frac{0.2}{0.5} = 0.4 = P(A)$ → independent.

1. $P(A) = 0.3$ , $P(B) = 0.6$ , $P(A|B) = 0.5$ . Find $P(A \cap B)$ and $P(A \cup B)$ .
1. Are A and B in Problem 1 independent?
1. Three machines produce items. Machine A: 50% of items, 2% defective. Machine B: 30%, 3% defective. Machine C: 20%, 5% defective. Find P(defective) and P(from A | defective).

Want to check your answers and get step-by-step solutions?

✓
$P(A|B) = \frac{P(A \cap B)}{P(B)}$ .
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Independent: $P(A \cap B) = P(A) \times P(B)$ .
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Mutually exclusive: $P(A \cap B) = 0$ .
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Mutually exclusive events are never independent (unless one has probability 0).

Tags:

#a-level#maths#statistics#probability#conditional

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