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A-Level Probability
What the A-Level syllabus expects for Probability, and how to practise it.
What the syllabus expects
- The addition and multiplication principles used in counting
- Permutations ⁿPᵣ and combinations ⁿCᵣ
- Arranging objects in a row or around a circle, allowing for repetition and restriction
- Adding and multiplying probabilities
- Mutually exclusive events and independent events
- Using outcome tables, Venn diagrams, tree diagrams and permutation/combination methods to find probabilities
- Computing conditional probabilities in simple situations
- Applying P(A') = 1−P(A), P(A∪B) = P(A)+P(B)−P(A∩B), and P(A|B) = P(A∩B)/P(B)
How it's examined
Questions on this topic most often ask you to find, determine, calculate, name. About 7% of the past-paper style questions in Rae's bank for this subject sit in this topic.
Worked examples
Example 1 (3 marks)
From the bag of 4 red and 6 blue balls, drawn one at a time without replacement until empty, find the probability that the first five draws contain exactly 2 blue balls, given they contain at least 3 red balls.
Show the worked answer
Bag has 4 red and 6 blue (10 total). Consider the first five draws; number of red among them can be 0..4. 'Exactly 2 blue' means 3 red and 2 blue. 'At least 3 red' means 3 or 4 red (4 is the maximum). P(exactly r red, 5-r blue in first five) = C(4,r)C(6,5-r) / C(10,5), with C(10,5) = 252. P(3 red, 2 blue) = C(4,3)C(6,2)/252 = 4*15/252 = 60/252. P(4 red, 1 blue) = C(4,4)C(6,1)/252 = 1*6/252 = 6/252. Required conditional probability = P(3 red) / [P(3 red) + P(4 red)] = 60 / (60 + 6) = 60/66 = 10/11.
Example 2 (5 marks)
Bella throws a biased six-faced die whose faces are labelled 1, 2, 3, 4, 5 and 6. Every odd-numbered face carries the same probability, and every even-numbered face carries the same probability, with the chance of an even number being twice the chance of an odd number. (a) Determine the probability of rolling a 1. [2] Bella now throws the die three times. (b) Determine the probability that she obtains two 1s and one 2. [3]
Show the worked answer
(a) Let odd faces each have probability p and even faces each 2p. Total: 3p+3(2p)=1 => 9p=1 => p=1/9. So P(1)=1/9. (b) P(1)=1/9, P(2)=2/9. Two 1s and one 2 in three throws: the single '2' can be in any of 3 positions, so P = 3 * (1/9)² * (2/9) = 3 * (1/81)*(2/9) = 6/729 = 2/243.
Example 3 (2 marks)
Kitty's 10 uniquely designed charms (5 small, 3 medium, 2 large) are set in a line along a photo frame's base with no two small charms adjacent. In how many ways?
Show the worked answer
There are 5 small charms and 5 non-small charms (3 medium + 2 large), all distinct. First arrange the 5 non-small charms in a row: 5! = 120 ways. This creates 6 gaps (including the two ends) into which small charms can go so that no two smalls are adjacent. Place the 5 distinct small charms into 5 of these 6 gaps, one per gap: P(6,5) = 6*5*4*3*2 = 720 ways. Total = 120 * 720 = 86400.
More worked questions on this topic
- Kitty owns 5 small, 3 medium and 2 large spherical charms, all uniquely designed. She sets all (2 marks)
- Aaron and Sandy do not get along and refuse to be in the same group. How many groupings of the (3 marks)
- Counting every 4-letter code that can be formed from the letters of BOOKKEEPER with no restrict (3 marks)
- With three receipts each showing one of 10 equally likely characters A–J, given that exactly on (3 marks)
- For the same free-throw game (Ben chosen with probability 0.7, single-attempt success 0.1 for B (3 marks)
- Using the wardrobes (A: T1, T2 formal, T3, T4 casual; B: P1, P2 formal, P3 casual), Without rep (3 marks)
- A 10-character code has its first 5 characters chosen from the letters {A,B,C,D,E,F,G,H} and it (5 marks)
- Eight tourists, being 2 married couples and 4 single travellers, share two taxis X and Y to the (5 marks)
- A pizzeria sells pizza by the slice, 8 slices making a whole pizza. (a) Three friends buy 4 Haw (4 marks)
More A-Level H2 Maths topics
Functions · Graphs and their transformations · Equations and inequalities · Sequences and series · Vectors in two and three dimensions: basic properties · Scalar and vector products · all of A-Level H2 Maths