GCSE · Combined events & tree diagrams

Probability without replacementExam-style questions

Why the second pick happens in a smaller, changed bag — so the total always drops by one but a colour's count drops only if that colour was taken — and how that turns "one of each", "at least one" and the reverse quadratic questions.

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17 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
There are $9$ pieces of fruit in a bowl.

$4$ are apples, $3$ are pears and $2$ are oranges.

Without looking, Sami takes three pieces of fruit from the bowl.

Work out the probability that, after this, the number of pears in the bowl is equal to the number of oranges in the bowl.

You must show all your working.

(5 marks)
Level 5 · 5 marksTry it
2
A hat contains $11$ tickets.

The tickets are numbered $10, 11, 12, \ldots, 20$

Ollie takes two of the tickets at random from the hat.

Ollie says,

"I am three times as likely to get an odd number when I add my two numbers as when I multiply them."

Is Ollie correct?

You must show how you get your answer.

(5 marks)
Level 5 · 5 marksTry it
3
A box contains $n$ light bulbs.

Exactly one of the bulbs is faulty.

Dan takes two of the bulbs at random from the box.

Show that the probability that one of the two bulbs Dan takes is the faulty bulb is $\dfrac{2}{n}$

(2 marks)
Level 5 · 2 marksTry it
4
Tess has a cool box of bottled water.

The bottles hold still water, sparkling water or flavoured water.

$x$ of the bottles are still water and $3$ of the bottles are sparkling water.

number of bottles of still water : number of bottles that are not still water $= 1 : 4$

Tess takes two of the bottles at random.

Find an expression, in terms of $x$ , for the probability that both of these bottles are the same type of water.

Give your answer in the form $\dfrac{ax^2 + bx + c}{dx^2 + ex}$ , where $a$ , $b$ , $c$ , $d$ and $e$ are integers.

(5 marks)
Level 5 · 5 marksTry it
5
A bag contains only red counters and blue counters.

There are $n$ red counters and 4 blue counters in the bag.

Nia takes a counter at random from the bag and does not replace it. She then takes a second counter at random.

The probability that both of Nia's counters are red is $\frac{1}{3}$ .

(a) Show that $n^2 - 5n - 6 = 0$ .

(b) Work out the probability that Nia's two counters are different colours.
Level 5 · 6 marksTry it