PLC Papers - WordPress.com · PiXL PLC 2017 Certification Probability of dependent events 1 Grade 5...

PLC Papers

Created For:

Probability Intervention

PiXL PLC 2017 Certification

Relative Frequency 1 Grade 4

Objective: Understand how relative expected frequencies relate to theoretical probability.

Question 1.

Jim, Hannah and Laura want to find out if a coin is biased.

They decide to toss the coin and count the number of times it lands on heads.

The table shows the number of trials each person completes and the number of times the coin lands on heads. (a) Complete the table to show the relative frequency for each experiment.

Give your answer as a decimal.

(3) (b) Which person did the most accurate experiment?

Explain your choice.

…………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………

(2) (c) Is the coin fair?

Explain your answer.

…………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………

(1)

(Total 6 marks)


Question 2. A bag contains 27 red beads and 18 blue beads.

I choose a bead from the bag at random, record the colour and replace it. (a) What is the probability that I will get a red bead?

..............................................

(2)

(b) If I repeat this experiment 80 times, how many times would I expect to get a red bead?

..............................................

(2)

(Total 4 marks)

Total /10


Probability of dependent events 1 Grade 5

Objective: To calculate the probability of dependent events using tree diagrams.

Question 1.

Grace has a packet of ten hyacinth bulbs. They all look the same.

Seven of the bulbs will produce pink flowers and three will produce blue flowers.

A bulb is taken at random and planted then a second bulb is taken at random and planted.

Calculate the probability that the two bulbs will produce at least one blue flower.

..............................................

(3)

(Total 3 marks)

Question 2

There are twenty students in a class. Fourteen of the students are girls.

Two students are randomly selected from the class, what is the probability that a boy and a girl are chosen?

..............................................

(2)

(Total 2 marks)


Question 3

A golfer observes that, when playing a particular hole at his local course, he hits a straight drive on 80% of

the occasions when the weather is not windy but only 30% of the occasions when the weather is windy.

Local records suggest that the weather is windy on 55% of all days.

a) Complete the tree diagram.

(2)

(b) Find the probability that the golfer hits a straight drive.

..............................................

(3)

(Total 5 marks)

Total /10


Unbiased samples 1 Grade 4

Objective: Understand that unbiased samples tend towards theoretical probability, with increasing sample size.

Question 1.

(a) A six sided die is thrown.

What is the probability that the die lands on the number 4?

..............................................

(1)

(b) The die is rolled 36 times.

The table shows the results of the experiment.

From the experiment, what is the probability that a 6 was rolled?

..............................................

(1)

(c) Do you think this is a fair die?

Give reasons for your answer.

…………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………

(1)

(d) What would you expect to happen if the die was rolled more times?

…………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………

(1)

(Total 4 marks)


Question 2. Sarah, George and Christopher want to find out if a coin is biased.






…………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………



…………………………………………………………………………………………………………………

…………………………………………………………………………………………………………………

(1)

(Total 6 marks)

Total /10


Venn Diagrams 1 Grade 4

Objective: To design and use Venn diagrams to calculate probability.

Question 1.

Here is a Venn diagram.

50 students are asked if they have a dog or cat.

• 29 have a dog. • 30 have a cat. • 8 have a dog, but not a cat.

(a) Complete the Venn diagram.

(3)

(b) A student is chosen at random.

What is the probability that this student has a cat and a dog?

..............................................

(1)

(Total 4 marks)


Question 2.

100 students were asked in a survey whether they used texts or social media.

• 35 students said they only use texts.

• 29 students said they only use social media.

• 21 students said they use both texts and social media.

(a) Put this information on the Venn diagram.

(1)

(b) One of the students in the survey is chosen at random.

What is the probability that this student uses social media?

..............................................

(2)

(Total 3 marks)


Question 3.

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

A = {multiples of 3}

A ∩ B = {3, 9, 12}

A ∪ B = {1, 2, 3, 6, 7, 9, 10, 11, 12, 14, 15}

Draw a Venn diagram for this information.

(3)

(Total 3 marks)

Total /10


Probability of independent events 1 Grade 5

Objective: To calculate the probability of independent events using tree diagrams.

Question 1.

Wendy goes to a fun fair. She has one go at Hoopla and one go on the Coconut shy. The probability that she wins at Hoopla is 0.4 whereas the probability that she wins on the Coconut shy is 0.3.

(a) Complete the probability tree diagram.

(2)

(b) Work out the probability that Wendy wins at Hoopla and also wins on the Coconut shy.

..............................................

(2)

(Total 4 marks)


Question 2

Lily and Anna take a test. The probability that Lily will pass the test is 0.6 The probability that Anna will pass the test is 0.8 (a) Work out the probability that both of these girls fail the test.

..............................................

(3)

(b) Work out the probability that both of these girls pass the test or that both of these girls fail the test.

..............................................

(3)

(Total 6 marks)

Total /10


Conditional Probability 1 Grade 7

Objective: Calculate and interpret conditional probabilities, using expected frequencies with two- way tables, tree diagrams and Venn diagrams

Question 1.

A bag of sweets contains 9 mints, 6 toffees and 5 sherbert lemons.

Helen takes 3 sweets at random from the bag.

Work out the probability that, of the three sweets Helen takes, exactly two will be the same flavour.

..............................................

(4)

(Total 4 marks)


Question 2.

A box contains 3 new batteries, 5 partly used batteries and 4 dead batteries.

Kelly takes two batteries at random.

Work out the probablitity that she picks two dead batteries.

..............................................

(3)

(Total 3 marks)


Question 3

Caleb either walks to school or travels by bus.

The probability that he walks to school is 0.75.

If he walks to school, the probability that he will be late is 0.3.

If he travels to school by bus, the probability that he will be late is 0.1.

Work out the probability that he will not be late.

..............................................

(3)

(Total 3 marks)

Total /10

PLC Papers

Created For:

Probability Intervention


Relative Frequency 1 Grade 4 SOLUTIONS

Objective: Understand how relative expected frequencies relate to theoretical probability.

Question 1.

Jim, Hannah and Laura want to find out if a coin is biased.




One mark for each correct relative frequency M3



Laura M1

Because she did the most trials M1

…………………………………………………………………………………………………………………



No, you would expect the relative frequency to be close to 0.5. M1

(Must compare relative frequency to 0.5).

…………………………………………………………………………………………………………………

(1)

(Total 6 marks)


Question 2. A bag contains 27 red beads and 18 blue beads.

I choose a bead from the bag at random, record the colour and replace it. (a) What is the probability that I will get a red bead?

27 + 18 = 45 M1 (45 seen as a denominator) 2745 =

35 A1 oe

..............................................

(2)

(b) If I repeat this experiment 80 times, how many times would I expect to get a red bead?

0.6 × 80 M1 ft (Allow their probability multiplied by 80)

= 48 times A1 ft

..............................................

(2)

(Total 4 marks)

Total /10


Probability of dependent events 1 Grade 5 SOLUTIONS

Objective: To calculate the probability of dependent events using tree diagrams.

Question 1.

Grace has a packet of ten hyacinth bulbs. They all look the same.

Seven of the bulbs will produce pink flowers and three will produce blue flowers.

A bulb is taken at random and planted then a second bulb is taken at random and planted.

Calculate the probability that the two bulbs will produce at least one blue flower.

310 x

29 or 310 x

79 or 710 x

39 M1

310 x

29 + 310 x

79 + 710 x

39 M1

4890 =

815 A1 oe

OR

710 x

69 = 4290 M1

1 - 4290 M1

4890 = 815 A1 oe

..............................................

(3)

(Total 3 marks)

Question 2

There are twenty students in a class. Fourteen of the students are girls.

Two students are randomly selected from the class, what is the probability that a boy and a girl are chosen?

1420 x

619 + 620 x

1419 M1

168380 =

4295 A1 oe

..............................................

(2)

(Total 2 marks)


Question 3

A golfer observes that, when playing a particular hole at his local course, he hits a straight drive on 80% of

the occasions when the weather is not windy but only 30% of the occasions when the weather is windy.

Local records suggest that the weather is windy on 55% of all days.

a) Complete the tree diagram.

Probabilities correctly place on diagram M2 (Allow M1 if at least 3 are correct)

(2)

b) Find the probability that the golfer hits a straight drive.

Not windy and straight 0.45 × 0.8 = 0.36 M1 Windy and straight 0.55 × 0.3 = 0.165 M1 Straight drive 0.36 + 0.165 = 0.525 A1 oe

..............................................

(3)

(Total 5 marks)

Total /10


Unbiased samples 1 Grade 4 SOLUTIONS

Objective: Understand that unbiased samples tend towards theoretical probability, with increasing sample size.

Question 1.

(a) A six sided die is thrown.

What is the probability that the die lands on the number 4?

16 A1

..............................................

(1)

(b) The die is rolled 36 times.

The table shows the results of the experiment.

From the experiment, what is the probability that a 6 was rolled?

736 A1

..............................................

(1)

(c) Do you think this is a fair die?

Give reasons for your answer.

Yes because you expect around 6 of each number if the die was fair. C1

…………………………………………………………………………………………………………………

(1)

(d) What would you expect to happen if the die was rolled more times?

The amount of each number would become closer to the theoretical probability. C1

…………………………………………………………………………………………………………………

(1)

(Total 4 marks)


Question 2. Sarah, George and Christopher want to find out if a coin is biased.


The table shows the number of trials each person completes and the number of times the coin lands on heads.

(a) Complete the table to show the relative frequency for each experiment.


One mark for each correct relative frequency M3



Christopher M1

Because he did the most trials M1

…………………………………………………………………………………………………………………



No, you would expect the relative frequency to be close to 0.5. M1

(Must compare relative frequency to 0.5).

…………………………………………………………………………………………………………………

(1)

(Total 6 marks)

Total /10


Venn Diagrams 1 Grade 4 SOLUTIONS

Objective: To design and use Venn diagrams to calculate probability.

Question 1.

Here is a Venn diagram.

50 students are asked if they have a dog or cat.

• 29 have a dog. • 30 have a cat. • 8 have a dog, but not a cat.

(a) Complete the Venn diagram. 29 – 8 for number with a cat and a dog M1 30 – 21 for number with only a cat M1 50 – (21 + 8 + 9) for number with no cats or dogs M1

(3)

(b) A student is chosen at random.

What is the probability that this student has a cat and a dog?

2150 A1

..............................................

(1)

(Total 4 marks)

21 9

12


Question 2.

100 students were asked in a survey whether they used texts or social media.

• 35 students said they only use texts.

• 29 students said they only use social media.

• 21 students said they use both texts and social media.

(a) Put this information on the Venn diagram.

All 3 entries correct A1

(1)

(b) One of the students in the survey is chosen at random.

What is the probability that this student uses social media?

21 + 29 = 50, the number of students who use social media M1

50100 =

12 A1

..............................................

(2)

(Total 3 marks)

35 29 21

15


Question 3.

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

A = {multiples of 3}

A ∩ B = {3, 9, 12}

A ∪ B = {1, 2, 3, 6, 7, 9, 10, 11, 12, 14, 15}

Draw a Venn diagram for this information.

A ∩ B in correct position M1

A ∪ B in correct position M1

Fully correct and labelled Venn diagram A1

(3)

(Total 3 marks)

Total /10

A B

3

9

12

6

15

1

2

4

5

7

8

10

11

13

14


Probability of independent events 1 Grade 5 SOLUTIONS

Objective: To calculate the probability of independent events using tree diagrams.

Question 1.

Wendy goes to a fun fair. She has one go at Hoopla and one go on the Coconut shy. The probability that she wins at Hoopla is 0.4 whereas the probability that she wins on the Coconut shy is 0.3.

(a) Complete the probability tree diagram.

B1 for 0.6 in correct position on tree diagram

B1 for 0.7, 0.3, 0.7 in correct positions on tree diagram

(2)

(b) Work out the probability that Wendy wins at Hoopla and also wins on the Coconut shy.

0.4 x 0.3 = (M1) 0.12 (A1)

.............................................

(2)

(Total 4 marks)


Question 2

Lily and Anna take a test. The probability that Lily will pass the test is 0.6 The probability that Anna will pass the test is 0.8 (a) Work out the probability that both of these girls fail the test.

1 – 0.6 = 0.4 1 – 0.8 = 0.2 B1 for 0.4 or 0.2 seen 0.4 x 0.2 M1 indication of correct branch formed on tree diagram or

otherwise, leading to 0.4 x 0.2

A1 0.08

..............................................

(3)

(b) Work out the probability that both of these girls pass the test or that both of these girls fail the test.

0.4 x 0.2 + 0.6 x 0.8 M1 for 0.6 x 0.8 or “0.4” x “0.2” M1 0.6 x 0.8 + “0.4” x “0.2” or “0.08” + “0.48”

A1 0.56

..............................................

(3)

(Total 6 marks)

Total /10


Conditional Probability 1 Grade 7 SOLUTIONS

Objective: Calculate and interpret conditional probabilities, using expected frequencies with two- way tables, tree diagrams and Venn diagrams

Question 1.

A bag of sweets contains 9 mints, 6 toffees and 5 sherbert lemons.

Helen takes 3 sweets at random from the bag.

Work out the probability that, of the three sweets Helen takes, exactly two will be the same flavour.

MMT 920 x

819 x 618 x 3 =

1895 MMS

920 x 819 x

518 x 3 = 319

TTM 620 x

519 x 918 x 3 =

976 Correct outcomes chosen M1

TTS 620 x

519 x 518 x 3 =

576 Multiplying each probability by 3 M1

SSM 520 x

419 x 918 x 3 =

338 Adding their probabilities M1

SST 520 x

419 x 618 x 3 =

119 Correct solution A1

P(two the same flavour) = 6395

Or

MMM 920 x

819 x 718 =

795 TTT

620 x 519 x

418 = 157 Correct outcomes chosen M1

SSS 520 x

419 x 318 =

1114 Multiplying probability of MTS by 6 M1

MTS 920 x

619 x 518 x 6 =

938 Subtracting their answer from 1 M1

Correct solution A1

P(two the same flavour) = 1 – 3295 =

6395

..............................................

(4)

(Total 4 marks)


Question 2.

A box contains 3 new batteries, 5 partly used batteries and 4 dead batteries.

Kelly takes two batteries at random.

Work out the probablitity that she picks two different types of batteries.

NP 312 x

511 = 544

ND 312 x

411 = 111

PN 512 x

311 = 544

PD 512 x

411 = 533 Multiplying each probability M1

DN 412 x

311 = 111 Adding their probabilities M1

DP 412 x

511 = 533 Correct solution A1

P(two different types) = 4766

Or

NN 312 x

211 = 122

PP 512 x

411 = 533 Multiplying probability of MTS by 6 M1

DD 412 x

311 = 111 Subtracting their answer from 1 M1

Correct solution A1

P(two the same flavour) = 1 – 1966 =

4766

..............................................

(3)

(Total 3 marks)


Question 3

Caleb either walks to school or travels by bus.

The probability that he walks to school is 0.75.

If he walks to school, the probability that he will be late is 0.3.

If he travels to school by bus, the probability that he will be late is 0.1.

Work out the probability that he will not be late.

0.75 x 0.7 = 0.525 or 0.25 x 0.9 = 0.225 M1

0.525 + 0.225 = M1

0.75 A1

..............................................

(3)

(Total 3 marks)

Total /10

PLC Papers - WordPress.com · PiXL PLC 2017 Certification Probability of dependent events 1 Grade 5...

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