A-level · Probability

Tree diagrams and sequential eventsExam-style questions

Using a tree diagram for two-stage experiments: multiply probabilities ALONG the branches of a path (AND), add across the paths that make up an event (OR), and use 1 − P(none) for "at least one". The crucial distinction between sampling WITH replacement (probabilities stay the same) and WITHOUT replacement (the totals, and so the later probabilities, change).

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12 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A shelf in Kim's room holds 5 novels and 3 biographies only. Kim takes one book at random from the shelf. She then takes a second book at random from the books that remain on the shelf.

In the tree diagram below, $r$ , $s$ , $t$ and $u$ are probabilities.(a) Complete the tree diagram by finding the values of $r$ , $s$ , $t$ and $u$

(3 marks)
(b) Find the probability that Kim takes two books of the same type.

(2 marks)
(c) Find the probability that Kim takes at least one biography.

(2 marks)
(d) Given that at least one of the books Kim takes is a biography, find the probability that the first book she takes is a biography.

(2 marks)

A library trolley holds $n$ returned books. Each book on the trolley is a novel, a biography or a poetry book, and the numbers of novels, biographies and poetry books are in the ratio $4 : 3 : 1$

A librarian takes four books from the trolley at random, one at a time and without replacement.

The probability that the first two books taken are novels and the next two are biographies is $\frac{1}{26}$
(e) Show that $n$ satisfies the equation

$11n^2 + bn + c = 0$

where $b$ and $c$ are integers to be found.

(4 marks)
Level 5 · 13 marksTry it