To solve all these questions, you will need to draw a Tree Diagram
Question 1
At the Card Shark Casino, a barrel contains twenty balls numbered from 1 to 20. Players are
invited to nominate any three numbers between 1 and 20. Two balls are then chosen from
the barrel at random. If the two chosen are among the nominated three, there is a prize.
Pete the punter nominates his three favourite numbers (all between 1 and 20). Determine
the probability that he wins a prize. (Draw a Tree Diagram first)
Let F = favourite number, N = non-favourite number.
Pr(𝑤𝑖𝑛𝑠) = Pr(𝐹𝐹)
=
3
2
×
20 19
=
3
190
Question 2
A bag contains 7 Collingwood keyrings and 4 Geelong keyrings. Two key rings are chosen at
random from the bag.
a. Draw a Tree Diagram to illustrate the possible outcomes and their probabilities.
CC
CG
GC
GG
b. Calculate the probability that two Collingwood keyrings are selected. Give your
answer as a fraction.
𝐏𝐫(𝑪𝑪) =
𝟕
𝟏𝟏
×
𝟔
𝟏𝟎
=
𝟐𝟏
𝟓𝟓
c. Calculate the probability that two of the same kind are selected. Give your answer
as a fraction.
Pr (two of same kind ) = 𝐏𝐫(𝑪𝑪) + 𝐏𝐫(𝑮𝑮) =
𝟐𝟏
𝟓𝟓
+
𝟒
𝟏𝟏
×
𝟑
𝟏𝟎
=
𝟐𝟕
𝟓𝟓
Question 3
Charlie and Ralph are two friends in the same Maths class. One day their Maths teacher
decides to give a chocolate frog to two students in the class. She writes the name of each of
her twenty students on a card and places the cards all in a box. She then chooses two
cards at random from the box.
a. What is the probability that Carl and Ralph will each get a chocolate frog?
Let C = Charlie gets a frog
Let R = Ralph gets a frog
Let S = someone else gets a frog
Pr (Both get a frog) = 𝐏𝐫(𝑪𝑹) + 𝐏𝐫⁡(𝑹𝑪)
=
𝟏
𝟐𝟎
×
𝟏
𝟏𝟗
⁡+
𝟏
𝟐𝟎
×
𝟏
𝟏𝟗
=
𝟏
𝟏𝟗𝟎
On another day, their Maths teacher decides to again award two chocolate frogs. Again she
chooses a card at random from the twenty cards in the box. However, this time, after
announcing the first winner, she places the card back in the box and then chooses another
card at random.
a. What is the probability that Ralph gets two chocolate frogs?
Pr (Ralph gets two frogs) = Pr (RR)
=
𝟏
𝟐𝟎
×
𝟏
𝟐𝟎
=
𝟏
𝟒𝟎𝟎
b. What is the probability that Charlie gets exacty one and Ralph does not?
= 𝐏𝐫(𝑪𝑺) + 𝐏𝐫(𝑺𝑪) =
=
𝟑𝟔
𝟒𝟎𝟎
=
𝟗
𝟏𝟎𝟎
𝟏 𝟏𝟖⁡ 𝟏𝟖 𝟏
×
+
×
𝟐𝟎 𝟐𝟎 𝟐𝟎 𝟐𝟎
c. What is the probability that they get at least one frog to share ?
= 𝟏 − 𝐏𝐫⁡(𝑺𝑺)
=𝟏−
=
𝟏𝟖 𝟏𝟖
×
𝟐𝟎 𝟐𝟎
𝟏𝟗
𝟏𝟎𝟎
Question 4
At a supermarket it has been found that the probability that a customer has to wait
more than 5 minutes for service is 0.26. Of three customers calculate the probability
that:
a. None of them has to wait more than 5 minutes. (Answer correct to three
decimal places)
M = waits more than 5
min
L = waits 5 min or less
Pr(none has to wait more than 5 min) = Pr (LLL)
= 𝟎. 𝟕𝟒 × 𝟎. 𝟕𝟒 × 𝟎. 𝟕𝟒 = 𝟎. 𝟒𝟎𝟓𝟐
b. Exactly two of them have to wait 5 minutes or more. (Answer correct to three
decimal places).
= 𝐏𝐫(𝑴𝑴𝑳) + 𝐏𝐫(𝑴𝑳𝑴) + 𝐏𝐫⁡(𝑳𝑴𝑴)
= 𝟑 × 𝟎. 𝟐𝟔 × 𝟎. 𝟐𝟔 × 𝟎. 𝟕𝟒
= 𝟎. 𝟏𝟓𝟎𝟏
Question 5
Two friends, Ivan and Ian are going to Etihad stadium to watch Collingwood’s first
game of 2014 against Freemantle. There is a probability of 0.4 that Ivan will arrive
late, and there is a probability of 0.7 that Ian will arrive late.
a. Calculate the probability that Ivan is on time and Ian is late.
𝐏𝐫(𝑰𝒗𝒂𝒏⁡𝒐𝒏⁡𝒕𝒊𝒎𝒆⁡𝒂𝒏𝒅⁡𝑰𝒂𝒏⁡𝒍𝒂𝒕𝒆)
= 𝟎. 𝟕 × 𝟎. 𝟒 = 𝟎. 𝟐𝟖
b. Calculate the probability that exactly one of them is late.
𝐏𝐫(𝒐𝒏𝒆⁡𝒊𝒔⁡𝒍𝒂𝒕𝒆) = 𝐏𝐫(𝑳𝑻) + 𝐏𝐫(𝑻𝑳)
= 𝟎. 𝟕 × 𝟎. 𝟔 + 𝟎. 𝟑 × 𝟎. 𝟒
= 𝟎. 𝟓𝟒
c. Calculate the probability that at least one of them is late.
𝑷𝒓(𝒂𝒕⁡𝒍𝒆𝒂𝒔𝒕⁡𝒐𝒏𝒆⁡𝒍𝒂𝒕𝒆) = ⁡𝟏 − 𝐏𝐫(𝑻𝑻)
= 𝟏 − 𝟎. 𝟑 × 𝟎. 𝟔
= 𝟎. 𝟖𝟐
ALTERNATIVE SOLUTION
𝐏𝐫(𝒂𝒕⁡𝒍𝒆𝒂𝒔𝒕⁡𝒐𝒏𝒆⁡𝒍𝒂𝒕𝒆) = 𝐏𝐫(𝑳𝑳) + 𝐏𝐫(𝑳𝑻) + 𝐏𝐫(𝑻𝑳)
= 𝟎. 𝟕 × 𝟎. 𝟒 + 𝟎. 𝟕 × 𝟎. 𝟔 + 𝟎. 𝟑 × 𝟎. 𝟒 = 𝟎. 𝟖𝟐
Question 6
Callum is playing in a chess tournament. He estimates that the probability that he will win
any individual game is 0.6, the probability that he will lose is 0.3 and the probability of a
draw is 0.1. If he plays two games, calculate:
a. The probability that he wins both.
𝐏𝐫(𝑾𝑾) = 𝟎. 𝟔 × 𝟎. 𝟔
= 𝟎. 𝟑𝟔
0.3
b. The probability that he gets the same result in both games.
𝐏𝐫(𝒔𝒂𝒎𝒆⁡𝒓𝒆𝒔𝒖𝒍𝒕⁡𝒊𝒏⁡𝒃𝒐𝒕𝒉⁡𝒈𝒂𝒎𝒆𝒔) = 𝐏𝐫(𝑾𝑾) + 𝐏𝐫(𝑫𝑫) + 𝐏𝐫⁡(𝑳𝑳)
= 𝟎. 𝟑𝟔 + 𝟎. 𝟏 × 𝟎. 𝟏 + 𝟎. 𝟑 × 𝟎. 𝟑
= 𝟎. 𝟒𝟓
c. The probability that he does not lose a game.
𝐏𝐫(𝒅𝒐𝒆𝒔⁡𝒏𝒐𝒕⁡𝒍𝒐𝒔𝒆⁡𝒂⁡𝒈𝒂𝒎𝒆) = 𝐏𝐫(𝑾𝑾) + 𝐏𝐫(𝑾𝑫) + 𝐏𝐫(𝑫𝑾) + 𝐏𝐫⁡(𝑫𝑫)
= 𝟎. 𝟑𝟔 + 𝟎. 𝟔 × 𝟎. 𝟏 + 𝟎. 𝟏 × 𝟎. 𝟔 + 𝟎. 𝟏 × 𝟎. 𝟏
= 𝟎. 𝟒𝟗