Unit 6 Section 3 : The Probability of Two Events

In this section we review the use of listings, tables and tree diagrams to calculate the probabilities of two events.

Example 1

An unbiased coin is tossed twice.

(a)

List all the possible outcomes.

The possible outcomes are:
H H
H T
T H
T T
So there are 4 possible outcomes that are all equally likely to occur as the coin is not biased.
(b)

What is the probability of obtaining two heads?

There is only one way of obtaining 2 heads, so:
p(2 heads) =
(c)

What is the probability of obtaining a head and a tail in any order?

There are two ways of obtaining a head and a tail, H T and T H, so:
p(a head and a tail) = =

Example 2

A red dice and a blue dice, both unbiased, are rolled at the same time. The scores on the two dice are then added together.

(a)

Use a table to show all the possible outcomes.

The following table shows all of the 36 possible outcomes:
Red Dice
Blue Dice
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
(b)

What is the probability of obtaining:

(i) a score of 5,
There are 4 ways of scoring 5, so:
p(5) = =
(ii) a score which is greater than 3,
There are 33 ways of obtaining a score greater than 3, so:
p(greater than 3) = =
(iii) a score which is an even number?
There are 18 ways of obtaining a score which is an even number, so:
p(even score) = =

Example 3

A card is taken at random from a pack of 52 playing cards, and then replaced. A second card is then drawn at random from the pack.
Use a tree diagram to determine the probability that:

We first note that, for a single card drawn from the pack,
p(Diamond) = = and p(not Diamond) = = .
We put these probabilities on the branches of the tree diagram below:
Note also that the probability for each combination, for example, two Diamonds, is determined by multiplying the probabilities along the branches.
(a)

both cards are Diamonds,

p(both Diamonds) =
(b)

at least one card is a Diamond,

p(at least one Diamond) = + + =
(c)

exactly one card is a Diamond,

p(exactly one Diamond) = + = =
(d)

neither card is a Diamond.

p(neither card a Diamond) =

Exercises

Question 1

The faces of an unbiased dice are painted so that 2 are red, 2 are blue and 2 are yellow. The dice is rolled twice. Three of the possible outcomes are listed below:

R R R B R Y
(a)

List all 9 possible outcomes.

(b)

What is the probability that:

(i) both faces are red,
(ii) both faces are the same colour,
(iii) the faces are of different colours?
Question 2

A spinner is marked with the letters A, B, C and D, so that each letter is equally likely to be obtained. The spinner is spun twice.

(a)

List the 16 possible outcomes.

(b)

What is the probability that:

(i) A is obtained twice,
(ii) A is obtained at least once,
(iii) both letters are the same,
(iv) the letter B is not obtained at all?
Question 3

Two fair dice are renumbered so that they have the following numbers on their faces:

1, 1, 2, 3, 4, 6
The dice are rolled at the same time, and their scores added together.
(a)

Draw a table to show the 36 possible outcomes.

First Dice
Second Dice
1 1 2 3 4 6
1
1
2
3
4
6
(b)

What is the probability that the total score is:

(i) 6,
(ii) 3,
(iii) greater than 10,
(iv) less than 5 ?
Question 4

A red spinner is marked with the numbers 1 to 4 and a blue spinner is marked with the numbers 1 to 5. On each spinner all the numbers are equally likely to be obtained. The two spinners are spun at the same time and the two scores are added together.

(a)

Draw a table to show the 20 possible outcomes.

Blue Spinner
Red Spinner
1 2 3 4 5
1
2
3
4
(b)

What is the probability that the total score on the two spinners is:

(i) an even number,
(ii) the number 7,
(iii) a number greater than 4,
(iv) a number less than 7 ?
Question 5

An unbiased dice is rolled and a fair coin is tossed at the same time.

(a)

Show all the possible outcomes in a table.

Dice
Coin
1 2 3 4 5 6
H
T
(b)

What is the probability of obtaining:

(i) a head and a 6,
(ii) a tail and an odd number,
(iii) a tail and a number less than 5 ?
Question 6
A coin is biased so that the probability of obtaining a head is
and the probability of obtaining a tail is .
(a)

Complete the following tree diagram to show the possible outcomes and probabilities if the coin is tossed twice.

× =
× =
× =
(b)

What is the probability of obtaining:

(i) 2 heads,
(ii) at least one head,
(iii) 2 tails,
(iv) exactly 1 tail ?
Question 7

An unbiased dice is rolled twice in a game. If a 1 or a 6 is obtained, you win a prize.

(a)

Complete the following tree diagram:

× =
× =
× =
(b)

What is the probability that a player wins:

(i) 2 prizes,
(ii) 1 prize,
(iii) at least 1 prize ?
Question 8

A card is taken at random from a pack of 52 playing cards. It is replaced and a second card is then taken at random from the pack. A card is said to be a 'Royal' card if it is a King, Queen or Jack.
Use a tree diagram to calculate the probability that:

(a)both cards are Royals,
(b)one card is a Royal,
(c)at least one card is a Royal,
(c)neither card is a Royal.
Question 9
The probability that a school bus is late on any day is .

Use a tree diagram to calculate the probability that on two consecutive days, the bus is:

(a)late twice,
(b)late once,
(c)never late.
Question 10
The probability that a piece of bread burns in a toaster is .
Two slices of bread are toasted, one after the other.
(a)

Use a tree diagram to calculate the probability that at least one of these slices of bread burns in the toaster.

p(at least one slice burnt) =
(b)

Extend your tree diagram to include toasting 3 slices, one at a time. Calculate the probability of at least one slice burning in the toaster.

p(at least one slice burnt) =
Question 11

A coin has two sides, heads and tails.

(a)

Chris is going to toss a coin. What is the probability that Chris will get heads? Write your answer as a fraction.

(b)

Sion is going to toss 2 coins. Copy and complete the following table to show the different results he could get.

First coinSecond coin
headsheads
(c)

Sion is going to toss 2 coins. What is the probability that he will get tails with both his coins? Write your answer as a fraction.

(d)

Dianne tossed one coin. She got tails. Dianne is going to toss another coin. What is the probability that she will get tails again with her next coin? Write your answer as a fraction.

Question 12

I have two fair dice. Each of the dice is numbered 1 to 6.

(a)
The probability that I will throw double 6 (both dice showing number 6) is
What is the probability that I will not throw double 6 ?
p(not double 6) =
(b)

I throw both dice and get double 6. Then I throw both dice again.
Which one answer from the list below describes the probability that I will throw double 6 this time?

less than
more than

I start again and throw both dice.

(c)

What is the probability that I will throw double 3 (both dice showing number 3) ?

p(double 3) =
(d)

What is the probability that I will throw a double? (It could be double 1 or double 2 or any other double.)

p(double) =
Question 13

On a road there are two sets of traffic lights. The traffic lights work independently.
For each set of traffic lights, the probability that a driver will have to stop is 0.7.

(a)

A woman is going to drive along the road.

(i)

What is the probability that she will have to stop at both sets of traffic lights?

0.7 × 0.7 = 0.49
(ii)

What is the probability that she will have to stop at only one of the two sets of traffic lights?

(0.7 × 0.3) + (0.3 × 0.7) = 0.42
(b)

In one year, a man drives 200 times along the road. Calculate an estimate of the number of times he drives through both sets of traffic lights without stopping.

p(drives through both sets of lights without stopping) = 0.3 × 0.3 = 0.09,
so the estimated number of times he goes through unstopped = 200 × 0.09 = 18.
Question 14

100 students were asked whether they studied French or German.

Results:
27 students studied both French and German.
(a)

What is the probability that a student chosen at random will study only one of the languages?

Note: Write the solution as a decimal or percentage

(b)

What is the probability that a student who is studying German is also studying French?

(c)

Two of the 100 students are chosen at random.
From the following calculations, choose one which shows the probability that both students study French and German.

Note: Choose a calculation by clicking on it.

×
+
+
×
×
Question 15

A company makes computer disks. It tested a random sample of the disks from a large batch. The company calculated the probability of any disk being defective as 0.025.
Glenda buys 2 disks.

(a)

Calculate the probability that both disks are defective.

(b)

Calculate the probability that only one of the disks is defective.

(c)

The company found 3 defective disks in the sample they tested.
How many disks were likely to have been tested?

Question 16
On a tropical island the probability of it raining on the first day of the rainy season is .
If it does not rain on the first day, the probability of it raining on the second day is .
If it rains on the first day, the probability of it raining more than 10 mm on the first day is .
If it rains on the second day but not on the first day, the probability of it raining more than 10 mm is .
You may find it helpful to copy and complete the tree diagram before answering the questions.
=
=
=
(a)

What is the probability that it rains more than 10 mm on the second day, and does not rain on the first?

× × =
(b)

What is the probability that it has rained by the end of the second day of the rainy season?

+ × = + = =
(c)

Is it possible to work out the probability of rain on both days from the information given?

Because we are not given the probability that it rains in the second day if it rains on the first.
Question 17

Pupils at a school invented a word game called Wordo. They tried it out with a large sample of people and found that the probability of winning Wordo was 0.6.
The pupils invented another word game, Lango. The same sample who had played Wordo then played Lango. The pupils drew this tree diagram to show the probabilities of winning.

(a)

What was the probability of someone from the sample winning Lango?

(0.6 × 0.8) + (0.4 × 0.55) = 0.7
(b)

What was the probability of someone from the sample winning only one of the two word games?

(0.6 × 0.2) + (0.4 × 0.55) = 0.34
(c)

The pupils also invented a dice game. They tried it out with the same sample of people who had already played Wordo and Lango.
The probability of winning the dice game was 0.9. This was found to be independent of the probabilities for Wordo and Lango.
Calculate the probability of someone from the sample winning two out of these three games.

(0.6 × 0.8 × 0.1) + (0.6 × 0.2 × 0.9) + (0.4 × 0.55 × 0.9) = 0.354
(d)

Calculate the probability of someone from the sample winning only one of these three games.

(0.6 × 0.2 × 0.1) + (0.4 × 0.55 × 0.1) + (0.4 × 0.45 × 0.9) = 0.196