12.06 Tree diagrams

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:

Three fair coins are tossed:

A coin is tossed twice.

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

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.

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

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.

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.

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:

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.

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.

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:

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.

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

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.

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.

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.