# Help with a level stats question

Question: When Charlie and Robin play a game, they toss a coin to see who goes first. If it is a head
then Charlie goes first; if it is a tail then Robin goes first. Charlie and Robin play four
games and each time the coin shows a head. Robin claims that the coin is definitely
biased. Calculate a relevant probability, and discuss the validity of Robin’s claim

Do I do P(H4T0) , P(H3T1) , P(H2T2) , P(H1T3)?
If so I am unsure where to go from there

Also do I times the P(H) by P(T) or add them?
(edited 7 months ago)
Original post by max33456789
Question: When Charlie and Robin play a game, they toss a coin to see who goes first. If it is a head
then Charlie goes first; if it is a tail then Robin goes first. Charlie and Robin play four
games and each time the coin shows a head. Robin claims that the coin is definitely
biased. Calculate a relevant probability, and discuss the validity of Robin’s claim

Do I do P(H4T0) , P(H3T1) , P(H2T2) , P(H1T3)?
If so I am unsure where to go from there

Also do I times the P(H) by P(T) or add them?

You could do it by drawing a simple tree or using the appropriate distribution to get the probability of 4 heads with a fair coin. Then think about what it represents/how often would this occur so is it definitely biased?
(edited 7 months ago)
Original post by max33456789
Question: When Charlie and Robin play a game, they toss a coin to see who goes first. If it is a head
then Charlie goes first; if it is a tail then Robin goes first. Charlie and Robin play four
games and each time the coin shows a head. Robin claims that the coin is definitely
biased. Calculate a relevant probability, and discuss the validity of Robin’s claim

Do I do P(H4T0) , P(H3T1) , P(H2T2) , P(H1T3)?
If so I am unsure where to go from there

Also do I times the P(H) by P(T) or add them?

You need to try and solve it ... P(4Heads, 0 Tails) would be a start ... you multiply the probs [GCSE work!!]
Original post by Muttley79
You need to try and solve it ... P(4Heads, 0 Tails) would be a start ... you multiply the probs [GCSE work!!]

Haha yes, stats isn't my strong point so I even struggle with the basics but i am getting there and doing practice like this
So would the next one be P(H3T1) which would then be 1/2^3 x 1/2?
Original post by mqb2766
You could do it by drawing a simple tree or using the appropriate distribution to get the probability of 4 heads with a fair coin. Then think about what it represents/how often would this occur so is it definitely biased?

So if I draw a tree diagram and get each the probabilities needed, how would I go about telling whether it's bias or not?
Original post by max33456789
So if I draw a tree diagram and get each the probabilities needed, how would I go about telling whether it's bias or not?

The only one you have to base a decision on is 4 heads. So what is that probability and what does it represent in terms of how often you would expect 4 heads to occur with a fair coin?
Original post by max33456789
Haha yes, stats isn't my strong point so I even struggle with the basics but i am getting there and doing practice like this
So would the next one be P(H3T1) which would then be 1/2^3 x 1/2?

This question is from an assessment ....

No, HHHT can happen in more than one way. How do you test for bias?
Original post by Muttley79
This question is from an assessment ....

No, HHHT can happen in more than one way. How do you test for bias?

Yea, it's on integral my teacher said they will mark it for me once i've done it.

Do I need to explore all probabilities using a tree diagram like mbqp2766 said?
Original post by mqb2766
The only one you have to base a decision on is 4 heads. So what is that probability and what does it represent in terms of how often you would expect 4 heads to occur with a fair coin?

Is the probability for 4 heads on a bias coin just 1/16 then? On a Fair coin would it be just 1/2?
Original post by max33456789
Is the probability for 4 heads on a bias coin just 1/16 then? On a Fair coin would it be just 1/2?

Cant help thinking you really need to hit the basic gcse stats stuff. For a fair coin, p(H)=1/2 so p(HHHH)=1/16. A biased coin could have any value for p(H) which is not 1/2 and so 4 heads would not be 1/16.
(edited 7 months ago)
Original post by max33456789
Yea, it's on integral my teacher said they will mark it for me once i've done it.

Do I need to explore all probabilities using a tree diagram like mbqp2766 said?

Note I didnt say you have to explore all probabilities - you certainly dont. But if youre confused about finding the probability of a single branch (adding, multiplying, ..) it can help you think about the problem.
Original post by max33456789
Yea, it's on integral my teacher said they will mark it for me once i've done it.

Do I need to explore all probabilities using a tree diagram like mbqp2766 said?

No, we are only interested in whether getting 4 Heads is a row tells us the coin is biased. I'd go back to your textbook and relearn GCSE probability.
Original post by mqb2766
You could do it by drawing a simple tree or using the appropriate distribution to get the probability of 4 heads with a fair coin. Then think about what it represents/how often would this occur so is it definitely biased?

It's a (very!) long time since I since I did the "statsy" part of probability, but without any a priori reason to expect a bias to heads v.s. tails it seems to me you "should" be looking at the two-tailed probability ("HHHH or TTTT").
Original post by DFranklin
It's a (very!) long time since I since I did the "statsy" part of probability, but without any a priori reason to expect a bias to heads v.s. tails it seems to me you "should" be looking at the two-tailed probability ("HHHH or TTTT").

Its a 3 marker (from a quick google) and Id guess theyre really testing whether its biased in favour of heads or not which would be a single tail. But a test of simply being biased would indeed be two tailed. For the 3 marks, Id guess its something like 1 mark for getting p(4H) for a fair coin, then a bit of a discussion about how likely/often to get as the first sequence (for a fair coin) and whether it can be concluded its definately biased.

It reads like a precursor to hypothesis testing question, so Id guess the 1/2 tailed test distinction probably isnt that important.
(edited 7 months ago)
Original post by mqb2766
Its a 3 marker (from a quick google) and Id guess theyre really testing whether its biased in favour of heads or not which would be a single tail. But a test of simply being biased would indeed be two tailed.

Yes those are the two options. The "rule of thumb" I use is which option would you use if you didn't already have the observed data? And I think here it's fairly clear you wouldn't specifically look for "biased in favour of heads".

[Yes, it's probably out of scope for what they were after (and fortunately it doesn't really make any difference to the conclusion), but I don't like questions/expected answers where knowing *more* about a topic becomes a disadvantage].
Original post by DFranklin
Yes those are the two options. The "rule of thumb" I use is which option would you use if you didn't already have the observed data? And I think here it's fairly clear you wouldn't specifically look for "biased in favour of heads".

[Yes, it's probably out of scope for what they were after (and fortunately it doesn't really make any difference to the conclusion), but I don't like questions/expected answers where knowing *more* about a topic becomes a disadvantage].

Pretty much agree with all of that. For me the key word(s) is definitely biased and obviously you cant say that, irrespective of whether is a 1/2 tail test. It reads like something to get the student thinking about making statistical decisions and whether something with a probability of 1/16 happens "normally". It obviously does ~6% of the time and so the type 1/2 errors etc with statistical decision making etc. Id expect that realising you get 4 heads ~6% of the time to be part of the answer, but much further analysis to be outside the scope of the question, but its simply a guess.
(edited 7 months ago)