Basic Probability (CIE A Level Maths: Probability & Statistics 1)

Exam Questions

3 hours32 questions
1
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3 marks

Write the following probability statements in words.
The first one has been done for you. 

(i)
straight P left parenthesis A intersection B right parenthesis “The probability that A has happened and B has happened.”
(ii)
straight P left parenthesis A union B right parenthesis
(iii)
straight P left parenthesis A to the power of apostrophe intersection B to the power of apostrophe right parenthesis
(iv)     

straight P left parenthesis left parenthesis A union B right parenthesis to the power of apostrophe right parenthesis

 

 

 

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2a
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2 marks

The heights, in cm, of 100 rockhopper penguins are recorded in the following table: 

Height, bold italic h  (cm)

Frequency

40 less or equal than h less than 45

12

45 less or equal than h less than 50

36

50 less or equal than h less than 55

42

55 less or equal than h less than 60

10

 

Write down the number of penguins that are 

(i)   at least 50 cm tall 

(ii)   between 45 and 55 cm tall.

2b
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2 marks

A rockhopper penguin is chosen at random.
Use your answers to part (a) to help determine the probability that the penguin is 

(i)   at least 50 cm tall 

(ii)   between 45 and 55 cm tall.

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3a
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2 marks

A and B are two events such that P left parenthesis A right parenthesis space equals space 0.3P left parenthesis B right parenthesis space equals space 0.8,  and P left parenthesis A space and space B right parenthesis space equals space 0.24 .
Use the formula P left parenthesis A space and space B right parenthesis space equals space P left parenthesis A right parenthesis cross times P left parenthesis B right parenthesis to determine whether A and B are independent or not.

3b
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2 marks

C and D are two events such that P left parenthesis C right parenthesis space equals space 0.42P left parenthesis D right parenthesis space equals space 0.51,  and P left parenthesis C space o r space D right parenthesis space equals space 0.95 .
Use the formula P left parenthesis C space o r space D right parenthesis space equals space P left parenthesis C right parenthesis plus P left parenthesis D right parenthesis to determine whether C and D are mutually exclusive or not.

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4a
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2 marks

You may use the formula P left parenthesis A space and space B right parenthesis space equals space P left parenthesis A right parenthesis cross times P left parenthesis B right parenthesis in this question. 

A fair spinner has five sections labelled ‘dog’, ‘cat’, ‘dog’, ‘dog’ and ‘rabbit’. 

The spinner is spun once.
Find the probability the spinner comes to rest on a section labelled 

   (i)   ‘dog’ 

   (ii)   ‘rabbit’.

4b
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3 marks

The spinner is spun twice.
Find the probability that

(i)

the spinner comes to rest on the section labelled ‘cat’ on both spins

(ii)

the spinner does not come to rest on a section labelled ‘dog’ on either spin.

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5a
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2 marks

A box of equally sized toy bricks contains 3 blue bricks and 2 red bricks. 

A random toy brick is removed from the box. Determine the probability that this brick is 

(i)
blue
(ii)
red.
5b
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3 marks

A second brick is taken from the box without the first one being replaced.
Copy and complete the tree diagram below that represents this situation.

q5b-easy-3-1-basic-probability-edexcel-a-level-maths-statistics

5c
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2 marks

Use your tree diagram from part (b) to determine the probability that both bricks taken from the box are blue.

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6a
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2 marks

A fair die is rolled and a fair coin is tossed.  

Complete the table below listing all possible outcomes. 

2-1-s-q--q6c--easy--cie-a-level-statistics

6b
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6 marks

The event A is the die landing on a multiple of three and the event B is the coin landing on heads. Write down:

(i)
straight P open parentheses B close parentheses
(ii)
straight P open parentheses straight A apostrophe close parentheses
(iii)
text P end text open parentheses A intersection B close parentheses
(iv)     

straight P open parentheses A union B close parentheses

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7a
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2 marks

A and B are two events such that  P left parenthesis A right parenthesis space equals space 0.25  and space P left parenthesis B right parenthesis space equals space 0.7.
Given that A and B are independent use the formula  P left parenthesis A space and space B right parenthesis space equals space P left parenthesis A right parenthesis space cross times space P left parenthesis B right parenthesis space to find  P left parenthesis A space and space B right parenthesis.

7b
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2 marks

C and D are two events such that  P left parenthesis C right parenthesis space equals space 0.4 space space and space space P left parenthesis D right parenthesis space equals space 0.5.
Given that C and D are mutually exclusive use the formula space P left parenthesis C space o r space D right parenthesis space equals space P left parenthesis C right parenthesis space plus space P left parenthesis D right parenthesis space space to find  P left parenthesis C space o r space D right parenthesis.

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8
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5 marks

The tree diagram below shows the probabilities that a pool player will successfully complete two successive trick shots.  Y indicates a successful trick shot, N indicates a failed trick shot.q8-easy-3-1-basic-probability-edexcel-a-level-maths-statistics

Use the probabilities given to determine the probability that the pool player

(i)
successfully completes both trick shots
(ii)
successfully completes exactly one trick shot
(iii)
successfully completes at least one trick shot.

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1a
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2 marks

A and B are two events such that  P left parenthesis A right parenthesis space equals space 0.35P left parenthesis B right parenthesis space equals space 0.25  and  P left parenthesis A union B right parenthesis equals 0.6.  State, with a reason, whether A and B  are mutually exclusive.

1b
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2 marks

C and D are two events such that  straight P left parenthesis C right parenthesis space equals space 0.2straight P left parenthesis D right parenthesis space equals space 0.4  and straight P left parenthesis C intersection D right parenthesis space equals space 0.18 . State, with a reason, whether C and D  are independent.

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2
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4 marks

The lengths, in cm, of 120 adult platypuses are recorded in the following table:

Length, l
(cm)

Frequency
(female)

Frequency
(male)

39 ≤ l < 42 14 0
42 ≤ l < 45 29 0
45 ≤ l < 48 12 7
48 ≤ l < 51 6 21
51 ≤ l < 54 3 19
54 ≤ l < 57 1 5
57 ≤ l < 60 0 2
60 ≤ l< 63 0 1


One platypus is chosen at random.  Find the probability that the platypus is:

(i)
male
(ii)
less than 51 cm long
(iii)
a male less than 45 cm long
(iv)
a female between 45 and 54 cm long.

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3
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4 marks

Two fair spinners each have three sectors numbered 1 to 3.  The two spinners are spun together and then the product of the numbers indicated on each spinner is recorded.

Find the probability of the product indicated by the spinners being

(i)
exactly 6
(ii)
less than 4
(iii)
an odd number.

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4a
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1 mark

A fair four-sided die and a fair six-sided die are rolled and the sum of the numbers on their uppermost faces is recorded. 

Complete the sample space below showing all possible outcomes.

2-1-sq---q4a--medium-cie-a-level-statistics

4b
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5 marks

An event A is ‘the sum on the two dice is more than 7’ and an event B is ‘scoring an odd number on one die and an even number on the other’. 

Find: 

(i)
straight P open parentheses A close parentheses
(ii)
straight P open parentheses B close parentheses
(iii)
straight P open parentheses A intersection B close parentheses
(iv)
straight P open parentheses A union B close parentheses
4c
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2 marks

Determine whether the events A and B are independent or not, justify your answer.

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5a
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5 marks

Two words are chosen at random from the word IMPOSSIBLE without replacement.

The event A is defined as ‘choosing two of the same letters’ and the event  B is defined as 'the first letter taken is a letter S’. 

Find:

(i)
straight P open parentheses A close parentheses
(ii)
straight P open parentheses B close parentheses
(iii)
straight P open parentheses A intersection B close parentheses
5b
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2 marks

Determine whether the events A and B are mutually exclusive, justify your answer.

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6
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5 marks

Angelo meets with his maths tutor, Miss Perry, once a week.  The probability that Miss Perry changes the time of the meeting is 0.2.  The probability that she is late is 0.15.  Otherwise she is on time.  For the next two weeks that they meet, find the probability that Angelo’s maths tutor:

(i)

is on time for both of the meetings

(ii)

changes the time of their meeting at least once

(iii)

is late for exactly one and changes the time of the other meeting.

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7a
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3 marks

A bag contains 13 yellow tokens and 7 green tokens. Two tokens are drawn from the bag without replacement.

Draw a tree diagram to represent this experiment.

7b
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3 marks

Find the probability that the two tokens drawn are the same colour.

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8a
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3 marks

In a game of Galactic Unicorns your Monocerian-class space frigate is attacking an evil Sargonian robot ship.  Your attack will either hit or miss the robot ship, the probability of hitting the ship is 0.7.  If you hit the robot ship then there is a probability of 0.8 that the ship will be destroyed, otherwise it will manage to escape.  If you miss the robot ship then there is a probability of 0.2 that it will manage to escape, otherwise it will surrender because it has witnessed the immense power of your rainbow lasers. 

Draw a tree diagram to represent this information.

8b
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3 marks

Find the probability that the robot ship

(i)
is destroyed
(ii)
manages to escape
(iii)
surrenders.
8c
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2 marks

Show that the events ‘you hit the robot ship’ and ‘the robot ship manages to escape’ are independent.

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1a
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2 marks

A spinner with five equal sections labelled 1, 2, 3, 4 and 5 is spun once, the number the spinner lands on is recorded and then it is spun again. The event  R  is ‘The product of the two numbers the spinner lands on is even’ and the event S is ‘The first number is greater than the second number’. 

Construct a sample space diagram to display the product of the two numbers when the spinner is spun twice.

1b
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4 marks

Find:

(i)
straight P open parentheses R close parentheses
(ii)
straight P open parentheses S apostrophe close parentheses
(iii)
straight P open parentheses R intersection S close parentheses
(iv)
straight P open parentheses R union S close parentheses
1c
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1 mark

Give an example to show that the events R and S are not mutually exclusive.

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2a
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2 marks

The histogram below shows the distribution of masses, in grams, of 80 newly hatched ducklings:q1-hard-3-1-basic-probability-edexcel-a-level-maths-statistics

Find the probability that a duckling chosen at random has a mass less than 54 space g.

2b
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3 marks

Estimate the probability that a duckling chosen at random has a mass greater than 53 space g.

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3
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4 marks

A game is played using a fair spinner with four sectors numbered 1 to 4, as well as a fair dice with its six sides numbered 1 to 6.  The spinner is spun and the dice is rolled, and the score in the game is defined as the (positive) difference between the two results.

Find the probability of the score in the game being

(i)
exactly 0
(ii)
3 or more
(iii)
a prime number.

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4
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4 marks

On any given day the probability that Tomás works on his poetry is 0.55, and the probability that he stops in at his local pub is 0.75.  Given that he always does at least one of those two things, find the probability that on a given day Tomás:

(i)
goes to the pub and works on his poetry 
(ii)
goes to the pub but doesn’t work on his poetry.

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5
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4 marks

A comma space B comma space C space and space D are four events such that P left parenthesis A right parenthesis space equals space 0.28 comma spacespace P left parenthesis B right parenthesis space equals space 0.33 comma spaceP left parenthesis C right parenthesis space equals space 0.43 and P left parenthesis D right parenthesis space equals space 0.25. 

(i)
Given that space B space and space C spaceare mutually exclusive, find P left parenthesis B space or space C right parenthesis.

(ii)
Given that A space and space D are independent, find P left parenthesis A space and space D right parenthesis.

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6a
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1 mark

Praewa plays sports every day after school.  She is four times more likely to choose football than tennis and three times more likely to choose basketball than tennis. She does not play any other sport. 

Show that the probability of Praewa choosing tennis is 1 over 8.

6b
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3 marks

Find the probability that Praewa chooses three different sports on the next three days in a row.

6c
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2 marks

On a five day week chosen at random, find the probability that Praewa chooses football every day.

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7a
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4 marks

A bag contains 12 red marbles, 7 green marbles and 1 black marble.  Two marbles are drawn from the bag without replacement.

Draw a tree diagram to represent this experiment.

7b
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3 marks

Find the probability that the two marbles drawn are not both the same colour.

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8a
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4 marks

In a game of Unicorns Versus Zombies your unicorn is attempting to use the magic of its horn to dispel a cloud of zombie apocalypse flies.  On the first attempt, the probability of the magic working is 0.7.  If the magic works, then there is a probability of 0.2 that the flies will be turned into glitter pixies and join your rainbow army, otherwise the flies will simply be dispelled.  If the magic does not work the first time you may try again, although the probability of your magic working the second time is only 0.6.  Similarly, if your magic does not work the second time you may try a third time, but on the third attempt the probability of your magic working is reduced to 0.5.  If your magic works on the second or third attempts the probabilities of dispelling the flies or turning them into glitter pixies are the same as for the magic working on the first attempt.  If your magic does not work on the third attempt, however, then your unicorn is turned into an evil zombie corn and joins the zombie horde.  In all cases, the game ends when either the flies are turned into glitter pixies, or the flies are dispelled, or your unicorn is turned into a zombie corn.

Draw a tree diagram to represent this information.

8b
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3 marks

Find the probability that

(i)
the flies are turned into glitter pixies
(ii)
the flies are dispelled
(iii)
your unicorn is turned into a zombiecorn.
8c
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1 mark

Give an example of two events in the game that are mutually exclusive.

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1a
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2 marks

A six -sided spinner with equal segments labelled 1, 1, 1, 2, 3, 3 is spun and a fair coin is tossed.  If the coin lands on heads the number recorded is one more than the number on the spinner. If the coin lands on tails the number on the spinner is doubled and recorded.  The event A is ‘the number recorded is a prime number’ and the event B is ‘the number recorded is not one of the original numbers on the spinner’. 

Show that the events A and B are mutually exclusive, justify your answer.

 

1b
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6 marks

One of the segments on the spinner showing the number 1 is changed to a number 4 and the experiment is repeated using the same rules as before.  

Find:

(i)
straight P open parentheses A close parentheses
(ii)
straight P open parentheses B close parentheses
(iii)
straight P open parentheses A apostrophe intersection space B close parentheses
(iv)
straight P open parentheses A union B close parentheses to the power of apostrophe
1c
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2 marks

After the change that has been made to the spinner, determine if the events A and B are still mutually exclusive. Justify your answer.

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2a
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2 marks

The histogram below shows the distribution of masses, in kg, of 100 newborn calves:q1-very-hard-3-1-basic-probability-edexcel-a-level-maths-statistics

Find the probability that a calf chosen at random has a mass greater than 21 space kg.

2b
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3 marks

Estimate the probability that a calf chosen at random has a mass less than 23.5 space kg.

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3
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4 marks

A game is played using a fair spinner with four sectors numbered 1 to 4, as well as a fair eight-sided dice with its sides numbered 1 to 8. The spinner is spun and the dice is rolled, and the score in the game is determined as follows:

  • if the number on the spinner is higher than the number on the dice, then the score is the sum of the two numbers;
  • if the number on the spinner is lower than the number on the dice, then the score is the (positive) difference of the two numbers;
  • if the numbers on the spinner and the dice are equal, then the score is the product of the two numbers.

Find the probability of the score in the game being

(i)
exactly 7
(ii)
10 or more
(iii)
a triangular number (1, 3, 6, 10, 15, 21, …).

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4
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5 marks

On any given day the probability that Björn has an avocado with his lunch is 0.39, while the probabilities of him having a bacon butty or a piece of carrot cake are 0.32 and 0.44 respectively.  He never has all three items on the same day, but the probability that he doesn’t have at least one of them is only 0.09.  Given that the probability of him having exactly two of the three items is the same regardless of which two items those are, find the probability that on a given day Björn has:

(i)
exactly one of the three items
(ii)
a bacon butty but not a piece of carrot cake.

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5
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5 marks

A comma space B space and space C are three events such that  straight P left parenthesis A right parenthesis space equals space 0.28 comma space straight P left parenthesis A union C right parenthesis space equals space 0.4 space space and straight P left parenthesis B intersection space C right parenthesis space equals space 0.02.

Given that A and C are mutually exclusive, and thatspace B spaceand C are independent, find straight P left parenthesis B right parenthesis comma straight P left parenthesis C right parenthesis and P left parenthesis C intersection B to the power of apostrophe right parenthesis.

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6a
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4 marks

Isabel has two boxes of chocolates.  Box A contains 4 caramels and 6 toffees, and boxspace B contains 5 caramels and x toffees.  Isabel chooses a chocolate at random from box A and moves it to box B.  Her friend then takes a chocolate at random from box B. 

Show that the probability Isabel’s friend takes a caramel is given by fraction numerator 27 over denominator 5 open parentheses x plus 6 close parentheses end fraction

6b
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4 marks

Given that the probability that Isabel’s friend takes a toffee is 11 over 20, find the value of x.

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7a
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4 marks

A bag contains 10 black tokens and 6 white tokens.  A token is drawn from the bag and its colour recorded, and then a fair coin is flipped.  If the coin lands on heads then a second token is drawn from the bag without replacing the first token.  If the coin lands on tails then the first token is replaced in the bag before a second token is drawn.

Draw a tree diagram to represent this experiment.

7b
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3 marks

Find the probability that the second token drawn is white.

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8a
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6 marks

The game Undead Redemption is played using three fair dice – a four-sided dice with the sides numbered 1 to 4, a six-sided dice with the sides numbered 1 to 6, and an eight-sided dice with the sides numbered 1 to 8.

In the game your character is battling a zombie.  The battle can last between one and three rounds, and it is resolved as follows:

  • In the first round, you and the zombie each roll the four-sided dice.  If your roll is greater than or equal to the zombie’s roll then the zombie is destroyed and the battle is over.  Otherwise your character is wounded and the battle goes on to the second round.
  • In the second round, you roll the four-sided dice and the zombie rolls the six-sided dice.  If your roll is greater than or equal to the zombie’s roll then the zombie is destroyed and the battle is over.  Otherwise your character is wounded again and the battle goes on to the third round.
  • In the third round, you roll the four-sided dice and the zombie rolls the eight-sided dice.  If your roll is greater than or equal to the zombie’s roll then the zombie is destroyed and the battle is over.  Otherwise your character is wounded for the third time and dies.

Draw a tree diagram to represent this information.

8b
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3 marks

Find the probability that

(i)

the zombie is destroyed

(ii)

your character dies

(iii)

your character is wounded one or more times but still manages to destroy the zombie

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