8 min readConcept sheet1-mark MCQ; a 2-mark or 3-mark question, sometimes with an unknown number of balls

Balls in a bag, numbered tokens and other draws

Count the balls of each colour and the total in the bag. If a probability is given, set up an equation to find the unknown number.

Ball from a bagP(colour) = balls of that colour ÷ total balls

All colours togetherThe probabilities of all the colours add up to 1

Unknown number of ballsWrite P as a fraction using x, equate it to the given value, solve

Numbered tokens or cardsList the numbers that fit, then count them

Worked example 1: marbles in a jar

Problem

A jar has 4 red, 9 green and 7 blue marbles. One marble is taken out at random. Find the probability that it is (i) green, (ii) not blue.

Working

Formula: P(E) = favourable outcomes ÷ total outcomes.
Total marbles = 4 + 9 + 7 = 20.
P(green) = 9/20.
P(blue) = 7/20, so P(not blue) = 1 − 7/20 = 13/20.
Check: not blue = red + green = 13 marbles, and 13/20 matches.

Answer

(i) 9/20, (ii) 13/20.

Worked example 2: finding an unknown number

Problem

A bag has 6 white balls and some black balls. The probability of drawing a black ball is 2/5. How many black balls are there?

Working

Let the number of black balls be x. Total = 6 + x.
Formula: P(black) = black balls ÷ total = x ÷ (6 + x).
So x ÷ (6 + x) = 2/5.
Cross-multiply: 5x = 2(6 + x) = 12 + 2x.
3x = 12, so x = 4.
Check: P(black) = 4/10 = 2/5.

Answer

There are 4 black balls.

Worked example 3: numbered tokens

Problem

A box has 60 tokens numbered 1 to 60. One token is drawn at random. Find the probability that its number is (i) a perfect square, (ii) a multiple of 6.

Working

Formula: P(E) = favourable outcomes ÷ total outcomes. Total = 60.
Perfect squares up to 60: 1, 4, 9, 16, 25, 36, 49 (7). P = 7/60.
Multiples of 6 up to 60: 6, 12, 18, and so on up to 60. There are 60 ÷ 6 = 10 of them.
P(multiple of 6) = 10/60 = 1/6.

Answer

(i) 7/60, (ii) 1/6.

Tips for bag and number questions

Add all the colours to get the total before you divide.
'Not red' means all the balls that are not red, or 1 − P(red).
If balls are added or removed, change both the count and the total.
For numbers, list them; do not guess. Check the first and the last number of the range.
1 is a perfect square but not a prime.

In real life

Picking a chit from a bowl to decide who goes first, or a raffle at a school fete, is like drawing a ball from a bag. Each chit has the same chance when all are folded alike and mixed well.

Exam angle

1 marks: “A bag has 4 red and 7 yellow balls. Find the probability of drawing a yellow ball.” Total = 11, so P(yellow) = 7/11.

2 marks: “A box has 20 balls. The probability of drawing a red ball is 3/10. How many balls are not red?” Red balls = (3/10) × 20 = 6. Balls that are not red = 20 − 6 = 14.

3 marks: “A bag has 4 red balls and some green balls. The probability of drawing a green ball is three times that of drawing a red ball. Find the number of green balls.” Let green = g. Then g ÷ (4 + g) = 3 × 4 ÷ (4 + g), so g = 12. Check: P(red) = 4/16 = 1/4 and P(green) = 12/16 = 3/4.

Remember it

Bag questions: total first, colour next, fraction last. Unknown balls? Call them x and cross-multiply.