AS & A Level · Practice questions · Conditional probability and independence

Shrink the sample space.

Six original Cambridge-style questions on conditional probability and independence, at Paper 5 standard.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Route wanted ÷ all routes allowed.

Formula: \(\text{P}(A \mid B) = \dfrac{\text{P}(A \cap B)}{\text{P}(B)}\).
Tables: the condition picks the row or column total to divide by.
Trees: find \(\text{P}(B)\) by adding every route through \(B\), then divide.
Independence: calculate \(\text{P}(A \cap B)\) and \(\text{P}(A)\,\text{P}(B)\); state equal or not.
Exclusive: \(\text{P}(A \cap B) = 0\), so (with non-zero probabilities) not independent.

Recall

[3 marks]

\(\text{P}(A) = 0.5\), \(\text{P}(B) = 0.6\) and \(\text{P}(A \cap B) = 0.2\). Find \(\text{P}(A \mid B)\) and \(\text{P}(B \mid A)\), and determine whether \(A\) and \(B\) are independent.

\(\text{P}(A \mid B) = \tfrac{0.2}{0.6} = \tfrac{1}{3}\); \(\text{P}(B \mid A) = \tfrac{0.2}{0.5} = 0.4\). ✓
\(\text{P}(A)\,\text{P}(B) = 0.3 \ne 0.2\), so not independent. ✓

Calculation

[4 marks]

200 students were asked how they travel to college.

	Car	Bus	Walk	Total
Year 12	30	45	25	100
Year 13	40	35	25	100
Total	70	80	50	200

A student is chosen at random. (a) Find P(bus | Year 12). (b) Find P(Year 13 | car). (c) Determine whether the events 'walks' and 'Year 12' are independent.

(a) \(\tfrac{45}{100} = 0.45\). ✓
(b) \(\tfrac{40}{70} = \tfrac{4}{7}\). ✓
(c) P(walk and Year 12) \(= \tfrac{25}{200} = 0.125\); P(walk) \(\times\) P(Year 12) \(= 0.25 \times 0.5 = 0.125\). Equal, so independent. ✓

Calculation

[4 marks]

2% of a population have a condition. A test gives a positive result for 95% of people with the condition and for 8% of people without it. A person is chosen at random and tested.

(a) Find the probability that the result is positive. (b) Given that the result is positive, find the probability that the person has the condition.

(a) \(0.02 \times 0.95 + 0.98 \times 0.08 = 0.019 + 0.0784 = 0.0974\). ✓
(b) \(\dfrac{0.019}{0.0974} = 0.195\) (3 s.f.). ✓

Analysis

[4 marks]

Two fair dice are thrown. \(A\) is 'the first die shows 3', \(B\) is 'the total is 7' and \(C\) is 'the total is 9'. Determine whether \(A\) and \(B\) are independent, and whether \(A\) and \(C\) are independent.

\(\text{P}(A) = \tfrac{1}{6}\), \(\text{P}(B) = \tfrac{6}{36} = \tfrac{1}{6}\), \(\text{P}(C) = \tfrac{4}{36} = \tfrac{1}{9}\). ✓
\(A \cap B\) is (3,4) only: \(\tfrac{1}{36} = \tfrac{1}{6} \times \tfrac{1}{6}\). Independent. ✓
\(A \cap C\) is (3,6) only: \(\tfrac{1}{36}\), but \(\tfrac{1}{6} \times \tfrac{1}{9} = \tfrac{1}{54}\). Not independent. ✓

Analysis

[4 marks]

A box contains 4 red and 6 blue pens. Two pens are taken at random without replacement.

(a) Find the probability that the second pen is blue. (b) Given that the second pen is blue, find the probability that the first pen was red.

(a) \(\tfrac{4}{10} \times \tfrac{6}{9} + \tfrac{6}{10} \times \tfrac{5}{9} = \tfrac{24}{90} + \tfrac{30}{90} = \tfrac{54}{90} = \tfrac{3}{5}\). ✓
(b) \(\dfrac{24/90}{54/90} = \tfrac{24}{54} = \tfrac{4}{9}\). ✓

Analysis

[6 marks]

A factory has two machines. Machine A makes 60% of the items and machine B makes the rest. 3% of items from A and 5% of items from B are faulty.

(a) An item is chosen at random. Find the probability that it is faulty. (b) Given that an item is faulty, find the probability that it was made by machine A. (c) Two items are chosen at random, independently, and both are faulty. Find the probability that both were made by machine B.

(a) \(0.6 \times 0.03 + 0.4 \times 0.05 = 0.018 + 0.02 = 0.038\). ✓
(b) \(\dfrac{0.018}{0.038} = 0.474\) (3 s.f.). ✓
(c) For one faulty item, P(B | faulty) \(= \dfrac{0.02}{0.038} = 0.5263\). ✓
The items are independent, so the probability is \(0.5263^2 = 0.277\) (3 s.f.). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.