Sec 2 · G3 (Express)

Probability

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1. Probability of single events

Term	Meaning	Example: a fair die
sample space	all the possible outcomes	$\{1, 2, 3, 4, 5, 6\}$
probability	$P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$	$P(\text{even}) = \dfrac{3}{6}$ , so $\dfrac{1}{2}$
impossible	$P(E) = 0$	$P(7) = 0$
certain	$P(E) = 1$	$P(\text{less than } 7) = 1$
not happening	$P(\text{not } E) = 1 - P(E)$	$P(\text{not } 6) = \dfrac{5}{6}$

A card from 1 to 20: P(a multiple of 3)

\displaystyle P = \dfrac{6}{20}

e.g.

\displaystyle \dfrac{6}{20}

\displaystyle = \dfrac{3}{10}

$3, 6, 9, 12, 15, 18$ : six of them

A bag of 4 red and some blue: P(red) is one third

\displaystyle \dfrac{4}{4 + b} = \dfrac{1}{3}

e.g.

\displaystyle \dfrac{4}{4 + b} = \dfrac{1}{3}

\displaystyle 12 = 4 + b

\displaystyle b = 8

📖 Worked example

Cards numbered $1$ to $15$ are shuffled and one is drawn at random. Find the probability that the number is a prime number.

Step 1 of 2

List the numbers that are a prime number

2, 3, 5, 7, 11, 13

That is

6

numbers out of

15

Change to the bag	Red balls	All the balls
$x$ red put in	$r + x$	$T + x$
$x$ blue taken out	$r$	$T - x$
$2x$ red and $x$ blue put in	$r + 2x$	$T + 3x$

📖 Worked example

A bag contains $4$ red balls and $6$ blue balls. $x$ more red balls are put into the bag. A ball is then taken at random. The probability that it is red is $\dfrac{3}{5}$ . Find the value of $x$ .

Step 1 of 3

The numbers of balls after the change, in terms of x

Red balls:

4 + x

; all the balls:

10 + x

📖 Worked example

The two-way table shows how some Sec 3 and Sec 4 students travel to school. One of the Sec 4 students is chosen at random. Find the probability that this Sec 4 student takes the bus.

	Bus	Walk	Total
Sec 3	12	8	20
Sec 4	9	11	20
Total	21	19	40

Step 1 of 3

Count the Sec 4 students who takes the bus

From the table:

9

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Single events questions

1. Probability of single events

Term	Meaning	Example: a fair die
sample space	all the possible outcomes	$\{1, 2, 3, 4, 5, 6\}$
probability	$P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$	$P(\text{even}) = \dfrac{3}{6}$ , so $\dfrac{1}{2}$
impossible	$P(E) = 0$	$P(7) = 0$
certain	$P(E) = 1$	$P(\text{less than } 7) = 1$
not happening	$P(\text{not } E) = 1 - P(E)$	$P(\text{not } 6) = \dfrac{5}{6}$

A card from 1 to 20: P(a multiple of 3)

\displaystyle P = \dfrac{6}{20}

e.g.

\displaystyle \dfrac{6}{20}

\displaystyle = \dfrac{3}{10}

$3, 6, 9, 12, 15, 18$ : six of them

A bag of 4 red and some blue: P(red) is one third

\displaystyle \dfrac{4}{4 + b} = \dfrac{1}{3}

e.g.

\displaystyle \dfrac{4}{4 + b} = \dfrac{1}{3}

\displaystyle 12 = 4 + b

\displaystyle b = 8

📖 Worked example

Cards numbered $1$ to $15$ are shuffled and one is drawn at random. Find the probability that the number is a prime number.

Step 1 of 2

List the numbers that are a prime number

2, 3, 5, 7, 11, 13

That is

6

numbers out of

15

Change to the bag	Red balls	All the balls
$x$ red put in	$r + x$	$T + x$
$x$ blue taken out	$r$	$T - x$
$2x$ red and $x$ blue put in	$r + 2x$	$T + 3x$

📖 Worked example

A bag contains $4$ red balls and $6$ blue balls. $x$ more red balls are put into the bag. A ball is then taken at random. The probability that it is red is $\dfrac{3}{5}$ . Find the value of $x$ .

Step 1 of 3

The numbers of balls after the change, in terms of x

Red balls:

4 + x

; all the balls:

10 + x

📖 Worked example

The two-way table shows how some Sec 3 and Sec 4 students travel to school. One of the Sec 4 students is chosen at random. Find the probability that this Sec 4 student takes the bus.

	Bus	Walk	Total
Sec 3	12	8	20
Sec 4	9	11	20
Total	21	19	40

Step 1 of 3

Count the Sec 4 students who takes the bus

From the table:

9

Practise this →