Unit 6 Section 2 : The Probability of a Single Event

In this section we consider the probabilities of equally likely events. When you roll a fair dice, each of the numbers 1 to 6 is equally likely to be on the uppermost face of the dice.

For equally likely events:
p(a particular outcome) =

Example 1

A card is taken at random from a full pack of 52 playing cards. What is the probability that it is:

Note: As each card is equally likely to be drawn from the pack there are 52 equally likely outcomes.
(a)

a red card,

There are 26 red cards in the pack, so:
p(red) = =
(b)

a 'Queen',

There are 4 Queens in the pack, so:
p(Queen) = =
(c)

a red 'Ace',

There are 2 red Aces in the pack, so:
p(red Ace) = =
(d)

the 'Seven of Hearts',

There is only one 7 of Hearts in the pack, so:
p(7 of Hearts) =
(e)

an even number?

There are 20 cards that have even numbers in the pack, so:
p(even number) = =

Example 2

A packet of sweets contains 18 red sweets, 12 green sweets and 10 yellow sweets. A sweet is taken at random from the packet. What is the probability that the sweet is:

The total number of sweets in the packet is 40, so there are 40 equally likely outcomes when one is taken at random.
(a)

red,

There are 18 red sweets in the packet, so:
p(red) = =
(b)

not green,

There are 28 sweets that are not green in the packet, so:
p(not green) = =
(c)

green or yellow ?

There are 22 sweets that are green or yellow in the packet, so:
p(green or yellow) = =

Example 3

You roll a fair dice 120 times. How many times would you expect to obtain:

(a)

a 6,

p(6) =
Expected number of 6s = × 120 = 20
(b)

an even score,

p(even score) = =
Expected number of even scores = × 120 = 60
(c)

a score of less than 5 ?

p(score less than 5) = =
Expected number of less than 5 = × 120 = 80

Exercises

Question 1

You roll a fair dice. What is the probability that you obtain:

(a)a five,
(b)a three,
(c)an even number,
(d)a multiple of 3,
(e)a number less than 6 ?
Question 2

A jar contains 9 red counters and 21 blue counters. A counter is taken at random from the jar. What is the probability that it is:

(a)red,
(b)blue,
(c)green?
Question 3

You take a card at random from a pack of 52 playing cards. What is the probability that the card is:

(a)a red King,
(b)a Queen or a King,
(c)a 5, 6 or 7,
(d)a Diamond,
(e)not a Club ?
Question 4

A jar contains 4 red balls, 3 green balls and 5 yellow balls. One ball is taken at random from the jar. What is the probability that it is:

(a)green,
(b)red,
(c)yellow,
(d)not red,
(e)yellow or red ?
Question 5

The faces of a regular tetrahedron are numbered 1 to 4. When it is rolled it lands face down on one of these numbers. What is the probability that this number is:

(a)2,
(b)3,
(c)1, 2 or 3,
(d)an even number ?
Question 6

A spinner is numbered as shown in the diagram. Each score is equally likely to occur. What is the probability of scoring:

(a)1,
(b)2,
(c)3,
(d)4,
(e)5,
(f)a number less than 6 ?
Question 7

You toss a fair coin 360 times.

(a)

How many times would you expect to obtain a head?

(b)

If you obtained 170 heads, would you think that the coin was biased?

No - the number of trials is not really large enough to have the probability settle down.
Question 8

A spinner has numbers 1 to 5, so that each number is equally likely to be scored. How many times would you expect to get a score of 5, if the spinner is spun:

(a)10 times,
(b)250 times,
(c)400 times ?
Question 9

A card is drawn at random from a pack of 52 playing cards, and then replaced. The process is repeated a total of 260 times. How many times would you expect the card drawn to be:

(a)a 7,
(b)a red Queen,
(c)a red card,
(d)a Heart,
(e)a card with an even number ?
Question 10

A six-sided spinner is shown in the diagram. It is spun 180 times.
How many times would you expect to obtain:

(a)a score of 1,
(b)a score less than 4,
(c)a score that is a prime number,
(d)a score of 4 ?
Question 11

Barry is doing an experiment. He drops 20 matchsticks at random onto a grid of parallel lines. Barry does the experiment 10 times and records his results. He wants to work out an estimate of probability.

Number of the 20 matchsticks that have fallen across a line
5764685357
(a)

Use Barry's data to work out the probability that a single matchstick when dropped will fall across one of the lines.

(5 + 7 + 6 + 4 + 6 + 8 + 5 + 3 + 5 + 7) ÷ 200 = = 0.28
(b)

Barry continues the experiment until he has dropped the 20 matchsticks 60 times.
About how many matchsticks in total would you expect to fall across one of the lines?

0.28 × 1200 = 336
Question 12

Les, Tom, Nia and Ann are in a singing competition. To decide the order in which they will sing all four names are put into a bag. Each name is taken out of the bag, one at a time, without looking.

(a)

Write down all the possible orders with Tom singing second.

(b)

In a different competition there are 8 singers.

The probability that Tom sings second is.

Work out the probability that Tom does not sing second.

Question 13

(a)

What is the probability of getting a 3 on this spinner?

(b)

Shade the following spinner so that the chance of getting a shaded section is double the chance of getting a non shaded section.

(c)

Shade the following spinner so that there is a 75% chance of getting a shaded section.

Question 14

Pat has 5 white beads and 1 black bead in her bag. She asks two friends about the probability of picking a black bead without looking in the bag.

Owen says: "It is because there are 5 white beads and 1 black bead."
Maria says: "It is because there are 6 beads and 1 is black."

(a)

Which of Pat's friends is correct?

Maria is correct. Owen is wrong because the selection is 1 black from a total of 6 in the bag.
(b)
Tracy has a different bag of black beads and white beads.
The probability of picking a black bead from Tracy's bag is .
What is the probability of picking a white bead from Tracy's bag?
(c)

How many black beads and how many white beads could be in Tracy's bag?

× n black beads and × n white beads
For any positive integer n e.g. 14 black and 12 white beads.
(d)
Peter has a different bag of black beads and white beads. Peter has more beads in total than Tracy.
The probability of picking a black bead from Peter's bag is also .
How many black beads and how many white beads could be in Peter's bag?
× m black beads and × m white beads
For any positive integer m greater than the n given in part (c) e.g. 30 black and 35 white beads.
Question 15

Brightlite company makes light bulbs. The state of the company's machines can be:

available for use and being used
or available for use but not needed
or broken down.
(a)

The table shows the probabilities of the state of the machines in July 1994. What is the missing probability?

State of machines: July 1994Probability
Available for use, being used
Available for use, not needed0.09
Broken down0.03
(b)

During another month the probability of a machine being available for use was 0.92. What was the probability of a machine being broken down?

(c)

Brightlite calculated the probabilities of a bulb failing within 1000 hours and within 2000 hours.
Copy and complete the table below to show the probabilities of a bulb still working at 1000 hours and at 2000 hours.

TimeFailedStill working
At 1000 hours0.07
At 2000 hours0.57
Question 16
A machine sells sweets in five different colours:
red, green, orange, yellow, purple.
You cannot choose which colour you get.
There are the same number of each colour in the machine.
Two boys want to buy a sweet each.
Ken does not like orange sweets or yellow sweets. Colin likes them all.
(a)

What is the probability that Ken will get a sweet that he likes?

p(Ken) =
(b)

What is the probability that Colin will get a sweet that he likes?

p(Colin) =
(c)

On the following scale draw an arrow to show the probability that Ken will get a sweet that he likes. Label the arrow 'Ken'.

Ken
(d)

Draw an arrow on the scale to show the probability that Colin will get a sweet that he likes. Label this arrow 'Colin'.

Colin
(e)

Mandy buys one sweet. The arrow on the following scale shows the probability that Mandy gets a sweet that she likes.
Write a sentence that could describe which sweets Mandy likes.

Mandy likes out of the 5 sweets.