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Date November 2021 Marks available 2 Reference code 21N.2.SL.TZ0.5
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Find Question number 5 Adapted from N/A

Question

Arianne plays a game of darts.

The distance that her darts land from the centre, O, of the board can be modelled by a normal distribution with mean 10cm and standard deviation 3cm.

Find the probability that

Each of Arianne’s throws is independent of her previous throws.

In a competition a player has three darts to throw on each turn. A point is scored if a player throws all three darts to land within a central area around O. When Arianne throws a dart the probability that it lands within this area is 0.8143.

In the competition Arianne has ten turns, each with three darts.

a dart lands less than 13cm from O.

[2]
a.i.

a dart lands more than 15cm from O.

[1]
a.ii.

Find the probability that Arianne throws two consecutive darts that land more than 15cm from O.

[2]
b.

Find the probability that Arianne does not score a point on a turn of three darts.

[2]
c.

Find the probability that Arianne scores at least 5 points in the competition.

[3]
d.i.

Find the probability that Arianne scores at least 5 points and less than 8 points.

[2]
d.ii.

Given that Arianne scores at least 5 points, find the probability that Arianne scores less than 8 points.

[2]
d.iii.

Markscheme

Let X be the random variable “distance from O”.

X~N10, 32

PX<13=0.841  0.841344            (M1)(A1)

 

[2 marks]

a.i.

PX>15=  0.0478  0.0477903            A1

 

[1 mark]

a.ii.

PX>15×PX>15            (M1)

=0.00228  0.00228391            A1

 

[2 marks]

b.

1-0.81433            (M1)

0.460  0.460050            A1

 

[2 marks]

c.

METHOD 1

let Y be the random variable “number of points scored”

evidence of use of binomial distribution           (M1)

Y~B10, 0.539949           (A1)

PY5= 0.717  0.716650.            A1

 

METHOD 2

let Q be the random variable “number of times a point is not scored”

evidence of use of binomial distribution           (M1)

Q~B10, 0.460050          (A1)

PQ5= 0.717  0.716650          A1

 

[3 marks]

d.i.

P5Y<8           (M1)

0.628  0.627788            A1


Note: Award M1 for a correct probability statement or indication of correct lower and upper bounds, 5 and 7.

[2 marks]

d.ii.

P5Y<8PY5 =0.6277880.716650           (M1)

0.876  0.876003            A1

 

[2 marks]

d.iii.

Examiners report

Candidates appeared well prepared for straightforward questions using the normal distribution. Most were able to earn marks through recognition of the compound probability event in part (b). The wording of the information in part (c) required careful thought. This acted as a clear discriminator, causing difficulty for most candidates. One common error in this part was the calculation 1-0.81432. Though at the end of the paper, it was pleasing to see many candidates identify the event in part (d) as binomial. A common error was the use of p=0.8413. It is recommended that candidates write down the distribution with associated parameters and support this with a probability statement. This will allow method and follow-through marks to be awarded in subsequent parts. Weaker candidates incorrectly used 8 as the upper bound. Those who made it to the end of the paper were often rewarded for correct division of their probabilities found in parts (d)(i)&(ii).

a.i.

Candidates appeared well prepared for straightforward questions using the normal distribution. Most were able to earn marks through recognition of the compound probability event in part (b). The wording of the information in part (c) required careful thought. This acted as a clear discriminator, causing difficulty for most candidates. One common error in this part was the calculation 1-0.81432. Though at the end of the paper, it was pleasing to see many candidates identify the event in part (d) as binomial. A common error was the use of p=0.8413. It is recommended that candidates write down the distribution with associated parameters and support this with a probability statement. This will allow method and follow-through marks to be awarded in subsequent parts. Weaker candidates incorrectly used 8 as the upper bound. Those who made it to the end of the paper were often rewarded for correct division of their probabilities found in parts (d)(i)&(ii).

a.ii.

Candidates appeared well prepared for straightforward questions using the normal distribution. Most were able to earn marks through recognition of the compound probability event in part (b). The wording of the information in part (c) required careful thought. This acted as a clear discriminator, causing difficulty for most candidates. One common error in this part was the calculation 1-0.81432. Though at the end of the paper, it was pleasing to see many candidates identify the event in part (d) as binomial. A common error was the use of p=0.8413. It is recommended that candidates write down the distribution with associated parameters and support this with a probability statement. This will allow method and follow-through marks to be awarded in subsequent parts. Weaker candidates incorrectly used 8 as the upper bound. Those who made it to the end of the paper were often rewarded for correct division of their probabilities found in parts (d)(i)&(ii).

b.

Candidates appeared well prepared for straightforward questions using the normal distribution. Most were able to earn marks through recognition of the compound probability event in part (b). The wording of the information in part (c) required careful thought. This acted as a clear discriminator, causing difficulty for most candidates. One common error in this part was the calculation 1-0.81432. Though at the end of the paper, it was pleasing to see many candidates identify the event in part (d) as binomial. A common error was the use of p=0.8413. It is recommended that candidates write down the distribution with associated parameters and support this with a probability statement. This will allow method and follow-through marks to be awarded in subsequent parts. Weaker candidates incorrectly used 8 as the upper bound. Those who made it to the end of the paper were often rewarded for correct division of their probabilities found in parts (d)(i)&(ii).

c.

Candidates appeared well prepared for straightforward questions using the normal distribution. Most were able to earn marks through recognition of the compound probability event in part (b). The wording of the information in part (c) required careful thought. This acted as a clear discriminator, causing difficulty for most candidates. One common error in this part was the calculation 1-0.81432. Though at the end of the paper, it was pleasing to see many candidates identify the event in part (d) as binomial. A common error was the use of p=0.8413. It is recommended that candidates write down the distribution with associated parameters and support this with a probability statement. This will allow method and follow-through marks to be awarded in subsequent parts. Weaker candidates incorrectly used 8 as the upper bound. Those who made it to the end of the paper were often rewarded for correct division of their probabilities found in parts (d)(i)&(ii).

d.i.

Candidates appeared well prepared for straightforward questions using the normal distribution. Most were able to earn marks through recognition of the compound probability event in part (b). The wording of the information in part (c) required careful thought. This acted as a clear discriminator, causing difficulty for most candidates. One common error in this part was the calculation 1-0.81432. Though at the end of the paper, it was pleasing to see many candidates identify the event in part (d) as binomial. A common error was the use of p=0.8413. It is recommended that candidates write down the distribution with associated parameters and support this with a probability statement. This will allow method and follow-through marks to be awarded in subsequent parts. Weaker candidates incorrectly used 8 as the upper bound. Those who made it to the end of the paper were often rewarded for correct division of their probabilities found in parts (d)(i)&(ii).

d.ii.

Candidates appeared well prepared for straightforward questions using the normal distribution. Most were able to earn marks through recognition of the compound probability event in part (b). The wording of the information in part (c) required careful thought. This acted as a clear discriminator, causing difficulty for most candidates. One common error in this part was the calculation 1-0.81432. Though at the end of the paper, it was pleasing to see many candidates identify the event in part (d) as binomial. A common error was the use of p=0.8413. It is recommended that candidates write down the distribution with associated parameters and support this with a probability statement. This will allow method and follow-through marks to be awarded in subsequent parts. Weaker candidates incorrectly used 8 as the upper bound. Those who made it to the end of the paper were often rewarded for correct division of their probabilities found in parts (d)(i)&(ii).

d.iii.

Syllabus sections

Topic 4—Statistics and probability » SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams
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Topic 4—Statistics and probability

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