You are Here: Home >< Maths

# STEP 2016 Solutions

Announcements Posted on
TSR's new app is coming! Sign up here to try it first >> 17-10-2016
1. 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 L-Tyrosine
10:
11: Solution by Mathematigien
12: Solution by Mathematigien
13: Solution by Mathemagicien
2. STEP I, Question 2:

PDF here - attached below as well - and image here:

Attached Images
3. Question 2 STEP I 2016.pdf (302.1 KB, 106 views)
4. STEP I, Question 10

PDF link here - image and PDF attached below:

Attached Images
5. Question 10 STEP I 2016.pdf (358.3 KB, 70 views)
6. (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:
Do you have the paper to upload for the rest of us?
7. STEP I, Question 3

8. (Original post by Exp!)
Do you have the paper to upload for the rest of us?
Insight314 will be uploading a scanned copy in an hour or so, all I've got are pictures.
9. (Original post by smartalan73)
STEP I, Question 3

Just attaching it in big size for you:

(side note, this was a gift question!)

(other side note, I forgot sin -4 was positive...)
10. (Original post by Zacken)
Insight314 will be uploading a scanned copy in an hour or so, all I've got are pictures.
11. (Original post by Exp!)
It's literally an hour's wait.
12. 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

13. (Original post by Zacken)
Just attaching it in big size for you:
Cheers, not that great with online stuff. I thought the question was really good too, reread it like 3 times to try and work out if there was some catch I wasn't getting.
14. q8.
15. (Original post by student0042)
Just going to attach this in a bigger size above. Thanks for the solution.
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 Mobile
17. (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
Questions like this illustrate why people should check out that statistics section more often.
18. (Original post by 13 1 20 8 42)
Questions like this illustrate why people should check out that statistics section more often.
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.
19. (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.
Pray for p.d.fs
20. 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.
21. (Original post by 13 1 20 8 42)
Pray for p.d.fs
May Siklos give us lots of nice questions on number theory and differential equations amen
22. (Original post by sweeneyrod)
May Siklos give us lots of nice questions on number theory and differential equations amen
Another prime number question would be noice

Posted from TSR Mobile

## Register

Thanks for posting! You just need to create an account in order to submit the post
1. this can't be left blank
2. this can't be left blank
3. this can't be left blank

6 characters or longer with both numbers and letters is safer

4. this can't be left empty
1. Oops, you need to agree to our Ts&Cs to register

Updated: July 25, 2016
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Today on TSR

### How does exam reform affect you?

From GCSE to A level, it's all changing

Poll
Useful resources