STEP I (paper here)
1: Solution by Student0042
2: Solution by Zacken
3: Solution by smartalan73
4: Solution by 13 1 20 8 42
5: Solution by KingRS
6: Solution by physicsmaths
7: Solution by KingRS
8: Solution by Student0042
9: Solution by rohanpritchard
10: Solution by Zacken
11: Solution by rohanpritchard (1) and Krollo (2)
12: Solution by Ecasx
13: Solution by Krollo
STEP II (paper here):
1: Solution riquix
2: Solution by Hauss
3: Solution by StrangeBanana
4: Solution by StrangeBanana
5: Solution by Hauss
6: Solution by Mathemagicien
7: Solution by KingRS (1) and krishdesai7 (2) and ThatPerson (3)
8: Solution by Mathemagicien
9: Solution by riquix (1) and StrangeBanana (2)
10: Solution by EwanClementson
11: Solution by FarhanHanif.93
12: Solution by EwanClementson
13: Solution by Mathemagicien
STEP III (paper here):
1: Solution by Alex_Aits (1) and Llewellyn (2)
2: Solution by riquix and Llewellyn (2)
3: Solution by Mathemagicien and Llewellyn (2)
4: Solution by Zacken
5: Solution by Farhan.Hanif93
6: Solution by YonRamiires
7:
8: Solution by student0042
9: Solution by LTyrosine
10:
11: Solution by Mathematigien
12: Solution by Mathematigien
13: Solution by Mathemagicien
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STEP 2016 Solutions
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 1
 15062016 09:34
Last edited by Zacken; 25062016 at 12:28.Post rating:3 
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 2
 15062016 09:38
STEP I, Question 2:
PDF here  attached below as well  and image here:
Post rating:2 
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 3
 15062016 09:40
STEP I, Question 10
PDF link here  image and PDF attached below:

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 4
 15062016 09:45
(Original post by Zacken)
STEP I
1:
2: Solution by Zacken
3:
4:
5:
6:
7:
8:
9:
10: Solution by Zacken
11:
12:
13: 
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 5
 15062016 09:48
STEP I, Question 3

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 6
 15062016 09:49
(Original post by Exp!)
Do you have the paper to upload for the rest of us? 
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 7
 15062016 09:51
(side note, this was a gift question!)
(other side note, I forgot sin 4 was positive...) 
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 8
 15062016 09:51
(Original post by Zacken)
Insight314 will be uploading a scanned copy in an hour or so, all I've got are pictures.Post rating:1 
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 9
 15062016 09:52
(Original post by Exp!)
Pictures are good enough for now please upload them 
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 10
 15062016 09:54
I'll put up my Q4 again, words hard to read but algebra should be clear (note slight issue with whether the circles ought to be arcs or full due to sign and stuff, probably a matter of a couple of marks)
edit: Note there is mistake on the line under "letting y = f(x)..", I do not include all of the RHS, but this is remedied straight after
Last edited by 13 1 20 8 42; 15062016 at 15:25. 
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 11
 15062016 09:55
(Original post by Zacken)
Just attaching it in big size for you: 
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 12
 15062016 10:06
q8.
Post rating:1 
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 13
 15062016 10:10
(Original post by student0042)

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 14
 15062016 10:16
STEP I 2015 Q12
(i) Alice tosses a fair coin two times and Bob tosses a fair coin three times. Find the probability that Bob obtains more heads than Alice.
(ii) Alice tosses a fair coin three times and Bob tosses a fair coin four times. Find the probability that Bob obtains more heads than Alice.
(iii) Alice and Bob both throw a fair coin n times. The probability that they obtain an equal number of heads is p1, and the probability that Bob obtains more heads is p2. If Bob now throws the coin n+1 times, find the probability (in terms of p1 and p2) that Bob throws more heads than Alice. Hence generalise your results to parts (i) and (ii).
Spoiler:Show
Let A be the number of heads that Alice obtains, and let B the number that Bob obtains.
(i) There are many ways of doing this. For example, calculate P( B>A  B = r) in the four cases r = 0,1,2,3. Then P(B>A) is the sum of these probabilities. The answer is 1/2.
(ii) Same as above, with just one more case to consider. The answer is again 1/2. See a pattern here...?
(iii) In the n, n+1 case, let B' be the number of heads that Bob obtains in his first n throws, and Bi be the number he obtains in his last ( n+1 'th) throw. So B = B' + Bi. Now P(B>A) = P(B' > A) + P( B' = A and Bi = 1) = p2 + 1/2 * p1. In the n, n case, P(A>B') = P(A<B') = p2 by symmetry. So the total probability is 1 = P(A>B') + P(A<B') + P(A=B) = 2p2 + p1. Therefore p2 + 1/2 * p1 = 1/2.
Generalisation: if Alice throws a fair coin n times, for any positive integer n, and Bob throws it n+1 times, the probability that Bob throws more heads than Alice is 1/2.
Posted from TSR MobilePost rating:1 
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 15
 15062016 10:22
(Original post by Ecasx)
STEP I 2015 Q12
(i) Alice tosses a fair coin two times and Bob tosses a fair coin three times. Find the probability that Bob obtains more heads than Alice.
(ii) Alice tosses a fair coin three times and Bob tosses a fair coin four times. Find the probability that Bob obtains more heads than Alice.
(iii) Alice and Bob both throw a fair coin n times. The probability that they obtain an equal number of heads is p1, and the probability that Bob obtains more heads is p2. If Bob now throws the coin n+1 times, find the probability (in terms of p1 and p2) that Bob throws more heads than Alice. Hence generalise your results to parts (i) and (ii).
Spoiler:Show
Let A be the number of heads that Alice obtains, and let B the number that Bob obtains.
(i) There are many ways of doing this. For example, calculate P( B>A  B = r) in the four cases r = 0,1,2,3. Then P(B>A) is the sum of these probabilities. The answer is 1/2.
(ii) Same as above, with just one more case to consider. The answer is again 1/2. See a pattern here...?
(iii) In the n, n+1 case, let B' be the number of heads that Bob obtains in his first n throws, and Bi be the number he obtains in his last ( n+1 'th) throw. So B = B' + Bi. Now P(B>A) = P(B' > A) + P( B' = A and Bi = 1) = p2 + 1/2 * p1. In the n, n case, P(A>B' = P(A<B' = p2 by symmetry. So the total probability is 1 = P(A>B' + P(A<B' + P(A=B) = 2p2 + p1. Therefore p2 + 1/2 * p1 = 1/2.
Generalisation: if Alice throws a fair coin n times, for any positive integer n, and Bob throws it n+1 times, the probability that Bob throws more heads than Alice is 1/2.
Posted from TSR Mobile 
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 16
 15062016 10:29
(Original post by 13 1 20 8 42)
Questions like this illustrate why people should check out that statistics section more often. 
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 17
 15062016 10:30
(Original post by sweeneyrod)
I really hope there are nice stats questions on II and III. Knowing my luck though there'll be one on some awful normal distribution thing, and one geometry one. 
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 18
 15062016 10:34
Q1. I feel like there was another part as it seems a bit small, but then again I only proved it for n=1 and said, do the same thing.
I'll let you blow it up again. 
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 19
 15062016 10:35
(Original post by 13 1 20 8 42)
Pray for p.d.fsPost rating:1 
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 20
 15062016 10:36
(Original post by sweeneyrod)
May Siklos give us lots of nice questions on number theory and differential equations amen
Posted from TSR MobilePost rating:1
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Updated: July 25, 2016
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