Which coin is more likely to be biased?

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Christian Lawson-Perfect	said	Ready to use	7 years, 9 months ago
Elliott Fletcher	said	Needs to be tested	7 years, 9 months ago
Chris Graham	said	Has some problems	7 years, 9 months ago

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Christian Lawson-Perfect 7 years, 9 months ago

Gave some feedback: Ready to use

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Christian Lawson-Perfect 7 years, 9 months ago

Saved a checkpoint:

Looks good!

Restore this checkpoint

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Elliott Fletcher 7 years, 9 months ago

Published this.

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Elliott Fletcher commented 7 years, 9 months ago

I have awarded part b) with a mark and tried to use careful wording in parts b) and c).

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Elliott Fletcher 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Elliott Fletcher 7 years, 9 months ago

Gave some feedback: Has some problems

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Chris Graham 7 years, 9 months ago

Gave some feedback: Has some problems

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Chris Graham commented 7 years, 9 months ago

I do like this question, however as it stands the marking is unsatisfactory. If there are no marks for b) then it is very hard to justify the mark for c), since it depends directly on the student's interpretation. I would prefer Christian's suggestion below of careful wording and keep the mark in part b). You need the same careful wording in c) too.

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Elliott Fletcher commented 7 years, 9 months ago

Thanks for your feedback Christian, i have changed part a to a "what's your opinion" question and awarded it no marks. Also i have added an extra part before this question asking the student to calculate the experimental probabilities of tossing heads for coin 1 and coin 2.

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Elliott Fletcher 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Christian Lawson-Perfect 7 years, 9 months ago

Gave some feedback: Has some problems

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Christian Lawson-Perfect 7 years, 9 months ago

Saved a checkpoint:

I just don't know if there should be any marks for part a! If it was just a "what's your opinion" question, with no marks, then part b might challenge the student's intuition if they picked coin 2.

Working out experimental probabilities for the two coins as a first step might help - P(heads) would be higher for coin 2, but the point is that the sample size is small.

Or, the wording needs to be much more careful. Something like, "for which coin is there more evidence of bias?"

Restore this checkpoint

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Elliott Fletcher 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Chris Graham 7 years, 9 months ago

Gave some feedback: Has some problems

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Chris Graham commented 7 years, 9 months ago

I got 5 heads and 5 tails for the "distractor", which makes part a) very straightforward. I suggest that the imbalance in proportion of heads/tails is greater for coin 2 that coin 1, as it was when you put it up on the screen in the learning lab.

I suggest the following changes to the wording:

In the statement on the first line: "It is known that one of the coins is biased".

Part a): "Based on the available results, ..."

In the advice:

"..., which would be the case for an unbiased coin". It is true in any case. The point being for an unbiased coin you would expect this to approach 0.5. You should explicitly state this.

"theoretical probability... should be..." it is a fact for an unbiased coin, no should about it.

"is greater than 50%" would cover 70, 51, 50.00001. You probably don't want to get bogged down in compound probability calculations, but "signficantly different" would be an improvement.

Same with the "is greater than 50%" in part b).

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Elliott Fletcher 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Elliott Fletcher 7 years, 9 months ago

Created this.

Name	Status	Author	Last Modified
Which coin is more likely to be biased?	Ready to use	Elliott Fletcher	20/11/2019 14:45
Which coin is more likely to be biased?	draft	Xiaodan Leng	11/07/2019 01:59

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Question statement

Two different coins are flipped a different number of times. It is known that one of the coins is biased.

The results for both coins are given in the table below.

Outcome	Coin 1	Coin 2
Heads	$\var{h1}$	$\var{h2}$
Tails	$\var{t1}$	$\var{t2}$

Give any introductory information the student needs.

			Name	Type	Generated Value

h1	integer	6500
t1	integer	3500
h2	integer	12
t2	integer	3
win	integer	10

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(6000..7000 #250)

JME function reference

Description

number of heads for coin 1.

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

6500

→ Used by:

Statement
Advice
"Part a)" → "Gap 0." - Minimum accepted value
"Part a)" → "Gap 0." - Maximum accepted value

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
Part b) Choose one from a list
1. Step Information only
Part c) Choose one from a list

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Calculate the experimental probability of tossing heads, $P(\text{heads})$ , for each coin. Enter your answers as fractions.

$P(\text{heads})\; \text{for Coin $1$} =$ Gap 0.

$P(\text{heads})\; \text{for Coin $2$} =$ Gap 1.

Advice

The results from the tosses of both coins are given in the table below.

Outcome	Coin 1	Coin 2
Heads	$\var{h1}$	$\var{h2}$
Tails	$\var{t1}$	$\var{t2}$

a)

We can calculate the number of tosses of each coin by adding together the number of heads and tails obtained for each coin.

Coin $1$ was tossed $\var{h1+t1}$ times.

Coin $2$ was tossed $\var{h2+t2}$ times.

This means that for Coin $1$ , the experimental probability of tossing heads is

$\begin{align} \displaystyle\frac{\var{h1}}{10000} &= \var{{h1}/10000}\\ &= \var{100*({h1}/10000)}\%. \end{align}$

Whereas for Coin $2$ , the experimental probability of tossing heads is

$\begin{align}\displaystyle\frac{\var{h2}}{15} &= \var{dpformat({h2}/15,2)} \; (\text{rounded to two decimal places})\\&= \var{dpformat(100*({h2}/15),0)}\%. \end{align}$

$\begin{align}\displaystyle\frac{\var{h2}}{15} &= \var{{h2}/15}\\&= \var{100*({h2}/15)}\%. \end{align}$

b)

We can see from part a) that the number of tosses of Coin $1$ is much larger than the number of tosses of Coin $2$ .

It is important to know that as the number of trials in an experiment gets very large the experimental probability tends towards the theoretical probability.

For an unbiased coin, the theoretical probability of tossing heads is $50\%$ so as the number of tosses gets very large we would expect the experimental probability of tossing heads to get closer to $50\%$ .

Therefore, as the number of tosses of Coin $1$ is very large and the experimental probability of tossing heads with this coin is significantly different from $50\%$ , then it is quite likely that Coin $1$ could be biased.

On the other hand, as the number of tosses for Coin $2$ is small, we cannot give an accurate opinion of whether this coin is biased or unbiased.

So, there is more evidence of bias for Coin $1$ than for Coin $2$ .

c)

As discussed in part b), there is more evidence of bias for Coin $1$ than for Coin $2$ .

Furthermore, we saw that the experimental probability of tossing heads with Coin $1$ is significantly different from $50\%$ , which means that the coin could be biased in favour of heads.

So, as we want the coin to land tails then we want to choose the coin that has the most reliable chance of landing tails-up.

Since there is less evidence of bias for Coin $2$ than for Coin $1$ , we should choose to use Coin $2$ in the bet as there is a more reliable chance of tossing tails with this coin.

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Information only
Choose one from a list
Choose one from a list		Hasn't run yet

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Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher commented 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Chris Graham 7 years, 9 months ago

Chris Graham commented 7 years, 9 months ago

Elliott Fletcher commented 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Chris Graham 7 years, 9 months ago

Chris Graham commented 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

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Which coin is more likely to be biased?

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher commented 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Chris Graham 7 years, 9 months ago

Chris Graham commented 7 years, 9 months ago

Elliott Fletcher commented 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Chris Graham 7 years, 9 months ago

Chris Graham commented 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`