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12.05 Tree diagrams and multi-stage events

Worksheet
Multistage events
1

Buzz can’t remember the combination for his lock, but he knows it is a three digit number and contains the digits 6, 8 and 9.

a

List all possible locker combinations that Buzz should try.

b

State the total number of possible outcomes.

c

If Buzz is correct that the combination includes 6, 8 and 9, find the probability that:

i

The combination starts with 6.

ii

The combination starts with 6 and ends with 8.

iii

The combination starts with 6 or ends with 9.

2

A number game uses a basket with 9 balls, all labelled with numbers from 1 to 9. 3 balls are drawn at random, without replacement. Find the probability that:

a

The ball labelled 3 is picked.

b

The ball labelled 3 is picked and the ball labelled 6 is also picked.

3

There are 4 green counters and 8 purple counters in a bag. Find the probability of choosing a green counter, not replacing it, then choosing a purple counter.

4

Mr and Mrs Smith are starting a family. They assume that having a girl is just as likely as having a boy.

a

Find the probability that the first child is a boy.

b

Find the probability that the first child is a boy and the second child is a girl.

c

Find the probability that the first two children are boys and the third child is a girl.

d

The Smiths have three children, all boys. Find the probability that the next child will be a girl.

e

Find the probability that in a four child family, the first three are boys and the fourth is a girl.

5

Consider the following four numbered cards:

Two of the cards are randomly chosen and the sum of their numbers is listed in the following sample space:

\left\{15, 10, 8, 11, 9, 4\right\}

a

Find the missing number on the fourth card.

b

If two cards are chosen at random, find the probability that the sum of their numbers is:

i

Even

ii

At least 10

Tables
6

Ben has 3 shirts, each in a different colour: crimson (C), pink (P) and white (W), and 4 ties, each in a different colour: blue (B), grey (G), red (R) and yellow (Y).

a

How many different combinations are possible?

Find the probability that he is wearing:

b

A pink shirt and yellow tie.

c

A pink shirt.

d

A pink or white shirt.

CPW
BC,BP,BW,B
GC,GP,GG,W
RC,RP,RW,R
YC,YP,YW,Y
7

A player rolls two dice and finds the sum of the numbers on the faces.

a

List the sample space for the sum of two dice.

b

Find the probability the dice will sum to five.

c

Find the probability the dice will sum to an odd number.

123456
1234567
2345678
3456789
45678910
567891011
6789101112
8

A player rolls two dice and finds the difference, that is, the largest number minus the smaller number.

a

List the sample space for the difference of two dice.

b

Find the probability the dice will have a difference of zero.

c

Find the probability the dice will have a difference of five.

123456
1012345
2101234
3210123
4321012
5432101
6543210
9

The following spinner is spun and a normal six-sided die is rolled:

WXYZ
11,W
22,Z
3
4
55,X
6
a

Complete the table above to represent all possible outcomes.

b

State the total number of possible outcomes.

c

Find the probability that the spinner lands on X and the dice rolls a prime number.

d

Find the probability that the spinner lands on W and the dice rolls a factor of 6.

e

Find the probability that the spinner doesn’t land on Z or the dice doesn't roll a multiple of 3.

10

Two dice are rolled, and the combination of numbers rolled on the dice is recorded.

a

Construct a table to represent the possible combinations.

b

Find the following probabilities for the two numbers rolled:

i

P(1 and 4)

ii

P(1 then 4)

iii

P(difference =4)

iv

P(product =12)

v

P(difference \leq 2)

vi

P(difference =3)

vii

P(product =20)

viii

P(difference\, \leq 1)

c

The numbers appearing on the uppermost faces are added. State whether the following are true.

i

A sum greater than 7 and a sum less than 7 are equally likely.

ii

A sum greater than 7 is more likely than a sum less than 7.

iii

A sum of 5 or 9 is more likely than a sum of 4 or 10.

iv

An even sum is more likely than an odd sum.

Tree diagrams
11

Hadyn is looking for a way to randomly choose a prime number between 1 and 20.

He first flips a coin to decide 0 or 1 and depending on the result he then rolls one of two four-sided dice, shown in the tree diagram. He arrives at his number by putting the coin number in front of the dice number.

a

List the sample space of numbers created.

b

Find the probability that the number will be 17.

c

Find the probability that the number will be less than 10.

d

Find the probability that the number will end in a 3.

12

An ice-cream shop randomly picks one of six flavours of ice-cream to sell at a discounted price each day. Three of the flavours are sorbet and the other three are gelato.

The three sorbet flavours are raspberry \left(R\right), lemon \left(L\right) and chocolate \left(C\right). The three gelato flavours are vanilla \left(V\right), mint \left(M\right) and chocolate \left(C\right).

a

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

b

Find the probability that the lemon sorbet is selected.

c

Find the probability that the flavour will be chocolate.

13

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.

14

Construct a tree diagram showing the following:

a

All possible outcomes of boys and girls that a couple with three children can possibly have.

b

All the ways the names of three candidates; Alvin, Sally and Peter, can be listed on a ballot paper.

15

A coin is tossed twice.

a

Construct a tree diagram showing the results of the given experiment.

b

Use the tree diagram to find the probability of getting:

i

Exactly 1 head.

ii

2 heads.

iii

No heads.

iv

1 head and 1 tail.

16

Three fair coins are tossed.

a

Construct the tree diagram for the experiment given.

b

Find the probability of obtaining:

i

At least one head.

ii

TTH in this order.

iii

THH in this order.

17

A die is rolled twice.

a

Construct a tree diagram showing all the possible results of the given experiment.

b

Use the tree diagram to find the probability of rolling:

i

A double 5.

ii

The same number twice.

iii

Two different numbers.

iv

Two odd numbers.

18

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.

a

Construct a tree diagram to indicate the possible pairs of traffic lights.

b

Find the probability that both sets of traffic lights will be yellow.

19

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.

20

Construct a probability tree showing all the ways a captain and a vice-captain can be selected from Matt, Rebecca, Helen and Chris.

Probability trees
21

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.

22

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

a

Construct a tree diagram for the seat Fiona receives and the seat her friend receives.

b

Find the probability that Fiona's friend has an aisle seat.

c

Find the probability of Fiona's friend receiving an aisle seat if Fiona has a window seat.

23

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

a

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

b

Find the probability that Hermione draws a black card and then a red card.

c

State whether the following events have an equal probability as 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 a black card and then another black card.

iv

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

v

Drawing at least one black card.

24

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

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

b

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

c

Find the probability that both jackets James selects are green.

25

An archer has three arrows that each have a probability of \dfrac{1}{5} of striking a target. If all three arrows are shot at a target:

a

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

b

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

c

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

d

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

26

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.

27

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.

28

Luke plays three tennis matches. In each match he has 60\% chance of winning.

a

Construct a probability tree showing all his possible outcomes and probabilities in these three matches.

b

Find the probability that he will win all his matches.

c

Find the probability that he will lose all his matches.

d

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

29

Each school day, Neil either rides his bike to school or walks. There is a 70\% chance Neil will ride his bike.

a

Construct a probability tree diagram showing Neil’s choices for three consecutive school days.

b

Find the probability that on the fourth day Neil walks to school.

c

Find the probability that Neil walks to school three days in a row.

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Outcomes

MS11-8

solves probability problems involving multistage events

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