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10.03 Tree diagrams

Worksheet
Tree diagrams
1

Three cards labeled 1, 2, 3 are placed face down on a table. Two of the cards are selected randomly to form a two-digit number. The possible outcomes are displayed in the following probability tree:

a

List the sample space of two digit numbers produced by this process.

b

Find the probability that 2 is a digit in the number.

c

Find the probability that the sum of the two selected cards is even.

d

Find the probability of forming a number greater than 40.

2

On the island of Timbuktoo, the probability that a set of traffic lights shows red, yellow or green is equally likely. Christa is travelling down a road where there are two sets of traffic lights.

What is the probability that both sets of traffic lights will be red?

3

Three fair coins are tossed:

a

Find the probability of obtaining at least 1 head.

b

Find the probability of obtaining TTH in this sequence.

c

Find the probability of obtaining THH in this sequence.

4

A coin is tossed twice.

a

Construct a tree diagram to identify the sample space of tossing a coin twice.

b

Use the tree diagram to find the probability of getting two tails.

c

Use the tree diagram to find the probability of getting at least one tail.

5

A bucket contains 5 green buttons and 7 black buttons. Two buttons are selected in succession from the bucket. The first button is replaced before the second button is selected.

a

Construct a tree diagram of this situation with the correct probability on each branch.

b

Find the probability of selecting two black buttons

6

A bag contains four marbles - red, green, blue and yellow. Beth randomly selects a marble, returns the marble to the bag and selects another marble.

a

Construct a tree diagram for the experiment given.

b

Find the probability of Beth selecting:

i

A blue and a yellow marble.

ii

A blue followed by a yellow marble.

iii

2 red marbles.

iv

2 marbles of the same colour.

v

2 marbles of different colours.

7

Three cards labelled 2, 3 and 4 are placed face down on a table. Two of the cards are selected randomly to form a two-digit number. The outcomes are displayed in the following probability tree diagram:

a

List the sample space of two digit numbers produced by this process.

b

Find the probability that:

i

2 appears as a digit in the number.

ii

The sum of the two selected cards is even.

iii

The number formed is greater than 40.

Probability trees
8

Han plays 3 tennis matches. In each match he has \dfrac{3}{5} chance of winning:

a

Find the probability that he will win all his matches.

b

Find the probability that he will lose all his matches.

c

Find the probability that he will win more matches than he loses.

9

Han owns four green ties and three blue ties. He selects one of the ties at random for himself and then another tie at random for his friend.

a

Write the probabilities for the outcomes on the edges of the probability tree diagram:

b

Calculate the probability that:

i

Han selects a blue tie for himself.

ii

Han selects two green ties.

10

Sally is drawing 2 cards from a deck of 52 cards. She draws the first card and checks whether it is red \left(R\right) or black \left(B\right). Without replacing her first card, she draws the second card and records its colour.

a

Write the probabilities for the outcomes on the edges of the probability tree diagram:

b

What is the probability that Sally draws a black card and then a red card?

c

State whether each of the following events has an equal probability to drawing a black then a red card:

i

Drawing a red card and then a black card.

ii

Drawing a red card and then another red card.

iii

Drawing at least one black card.

iv

Drawing a black card and then another black card.

v

Drawing one black card and one red card in any order.

11

The proportion of scholarship recipients at a particular university is \dfrac{7}{10}. The number of students at the university is so large that even if a student is removed, we can say that the proportion of scholarship recipients remains the same. If three students are selected at random:

a

Construct a probability tree showing all the possible combinations of recipients and nonrecipients.

b

Find the probability that at least one of the students is a scholarship recipient.

c

Find the probability that at least one of the students is a nonrecipient.

d

Find the probability there is at least one recipient and one nonrecipient in the selection.

12

Bart is purchasing a plane ticket to Adelaide. He notices there are only 4 seats remaining, 1 of them is a window seat \left(W\right) and the other 3 are aisle seats \left(A\right). His friend gets on the computer and purchases a ticket immediately after. The seats are randomly allocated at the time of purchase.

a

Write the probabilities for the outcomes on the edges of the probability tree diagram for the seat Bart receives and the seat his friend receives:

b

Find the probability that:

i

Bart's friend has an aisle seat.

ii

Bart's friend receives an aisle seat if Bart has a window seat.

13

A fair coin is tossed and then the following spinner is spun:

a

Construct a probability tree representing the situation.

b

Find the probability of getting a tail and then a yellow.

c

Find the probability of getting a tail, a yellow, or both.

d

Find the probability of getting a head and not getting a red.

e

Find the probability of not getting a head or a red.

14

James owns four green jackets and three blue jackets. He selects one of the jackets at random for himself and then another jacket at random for his friend:

a

Find the probability that James selects a blue jacket for himself.

b

Find the probability that both jackets James selects are green.

15

An archer has three arrows that each have a probability of \dfrac{1}{5} of striking a target. The tree diagram shows all the possible outcomes and probabilites:

a

Find the probability that all three arrows will hit the target.

b

Find the probability that at least one arrow will miss the target.

c

Find the probability that at least one arrow will hit the target.

16

An ice-cream shop offers one flavour of ice-cream at a discounted price each day. There are 6 flavours for the owner to choose from, that could be discounted. 3 of the flavours are sorbet and the other 3 are gelato. The three sorbet flavours are raspberry (R), lemon (L) and chocolate (C). The three gelato flavours are vanilla (V), mint (M) and chocolate (C). Each decision has equal probability.

a

Write the sample space for which ice-cream type and flavour may be chosen.

b

Find the probability that the lemon sorbet is selected.

c

Determine the probability that the flavour will be chocolate.

17

Two marbles are randomly drawn without replacement from a bag containing 1 blue, 2 red and 3 yellow marbles.

a

Construct a tree diagram to show the sample space.

b

Find the probability of drawing the following:

i

A blue marble and a yellow marble, in that order.

ii

A red marble and a blue marble, in that order.

iii

2 red marbles.

iv

No yellow marbles.

v

2 blue marbles.

vi

A yellow marble and a red marble, in that order.

vii

A yellow and a red marble, in any order.

18

Sophia has three races to swim at her school swimming carnival. The probability that she wins a particular race is dependent on whether she won the previous races, as summarised below:

  • The chance she wins the first race is 0.7.

  • If she wins the first race the chance of winning the second is 0.8.

  • If she loses the first race then the chance of winning the second is 0.4.

  • If she wins the first two then the chance of winning the third race is 0.9.

  • If she lost the first two then her chance of losing the third race is 0.9.

  • If she won only one of the first two races, then the chance of winning the third is 0.6.

a

Construct a probability tree to represent all outcomes in this situation.

b

Calculate the probability Sophia won all three races correct to three decimal places.

c

Calculate the probability Sophia won the third race, correct to three decimal places.

19

For breakfast, Maria has something to eat and drinks a hot drink. She will either eat toast or cereal and will drink tea or Milo.

  • The chance of Maria making toast is 0.7.

  • The chance of Maria drinking Milo is 0.4.

a

Construct a tree diagram illustrating all possible combinations of food and drink Maria can have for breakfast and their associated probabilities.

b

Find the probability Maria drinks tea and eats toast.

c

Find the probability Maria drinks tea or eats toast.

20

One cube has 4 red faces and 2 blue faces, another cube has 3 red faces and 3 blue faces, and the final cube has 2 red faces and 4 blue faces. The three cubes are rolled like dice.

a

Construct a probability tree diagram that shows all possible outcomes and probabilities.

b

Find the probability that three red faces are rolled.

c

Find the probability that more red faces than blue faces are rolled.

d

Find the probability that only one cube rolls a blue face.

21

A coin is tossed, then the spinner shown is spun and either lands on A, B or C.

Segment B is \dfrac{1}{8} of the entire cirle.

a

Construct a probability tree diagram showing all possible outcomes and probabilities.

b

Find the probability of landing on tails and the spinner landing on A.

c

Find the probability of landing on tails, or the spinner landing on A, or both.

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Outcomes

VCMSP347

Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence.

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