# Probability and ratio

1. Nov 15, 2012

### h0llow

1. The problem statement, all variables and given/known data
Game of Head & Tails between 2 people. The first one to 8 is the winner. If the game is interrupted: when one person has 6 wins and the other has 4 wins, how should the prize money be split?

2. The attempt at a solution

The obvious answer (at first) i got is 3:2
But it just seems too easy.. i think the part 8 wins should fit in somewhere but i have no idea how =/.

2. Nov 15, 2012

### tiny-tim

welcome to pf!

hi h0llow! welcome to pf!
if the game was interrupted just long enough to place bets on it,

what should the odds be?

3. Nov 15, 2012

### h0llow

assuming it was 100\$... person with 6 wins would have 75% chance of winning? and the other would have 50% chance..but if add 75+50 it adds up more than 100 =/

4. Nov 15, 2012

### tiny-tim

you mean, because 6 is 75% of 8, and 4 is 50% of 8 ?

no!

work out the probability that (starting from 0-0) there are 2 heads before there are 4 tails

5. Nov 15, 2012

### h0llow

huh 0_o? can u rephrase that please =/?

6. Nov 15, 2012

### tiny-tim

if you keep tossing a coin,

work out the probability that (starting from 0-0)

the 2nd head comes before the 4th tail​

(or, if you prefer the opposite, that the 4th tail comes when there are only 0 heads or 1 head)

7. Nov 15, 2012

### h0llow

considering there's a 50% chance..during each toss.. 2nd head has 25% chance? and 4th tail has 6.25% chance?

8. Nov 15, 2012

### Ray Vickson

You need to account for the fact that there might not be a second head at all. Just "guessing" does not work in problems of this type; you need to write things down in detail.

Also: the answer will depend quite a bit on whether the two-heads player tosses first or second.

RGV

9. Nov 15, 2012

### tiny-tim

sorry, but that's no way to solve this

try these

what is the probability that a coin tossed 4 times will come out all tails?

what is the probability that a coin tossed 5 times will come out 1 head and 4 tails, and that the head isn't the last one?​

10. Nov 15, 2012

### h0llow

(I) 0.5 X 0.5 X 0.5 X 0.5 = 6.25% (1/2 chance it is tail)
(ii) 3.125%?? (im not sure how exactly to do it)

11. Nov 15, 2012

### tiny-tim

yes
write out all the possible combinations

12. Nov 15, 2012

### h0llow

0.5 x 0.5 x 0.5 x 0.5 x 0.5??

13. Nov 15, 2012

### Ray Vickson

Don't keep guessing; that is no way to learn. As I said before: write things out in detail. By this, I mean: start enumerating some of the possibilities, until you see a pattern starting to emerge.

The outcome strings can be lists of H and T. If the two-head player (A) starts, then passes the coin to player B, etc., a string such as HHTHT means A gets H on the first toss, then B gets H on the second toss (that is, on B's first toss), the A gets T on the third toss (that is, on A's second toss), etc. Player A wins on his second toss if the outcomes are HHH or HTH. Player A wins on his third toss if the outcomes are HHTHH, HHTTH, HTTHH, HTTTH, THHHH, THHTH, TTHHH, TTHTH, etc. You need to figure out all the probabilities and add them up.

Then, you need to repeat this type of analysis if A goes second.

RGV

14. Nov 15, 2012

### h0llow

alright thank you

15. Nov 15, 2012

### h0llow

so i should write out all the 120 possibilities? (5!)

16. Nov 15, 2012

### h0llow

nvm.. i got it...can u tell me if i am right though?

TT THHT HHHH
HTT HTHT HTHHH
HHTT THHHT THHHH
HHHTT THTH HHTHH
THT HHTHT HHHTH

= 15 possibilities

chance player A (T) wins = 10/15 = 66.67%
chance player B (H) wins = 5/15 = 33.33%

am i right?

17. Nov 15, 2012

### h0llow

nvm.. i got it...can u tell me if i am right though?

TT THHT HHHH
HTT HTHT HTHHH
HHTT THHHT THHHH
HHHTT THTH HHTHH
THT HHTHT HHHTH

= 15 possibilities

chance player A (T) wins = 10/15 = 66.67%
chance player B (H) wins = 5/15 = 33.33%

am i right?

18. Nov 15, 2012

### tiny-tim

hi h0llow!
that looks right
no, because not all of them have equal probability

for example, the first one, TT actually is 8 times as likely as THHHH,

because it includes TTHHH TTHHT TTHTH TTHTT THTHH THTHT THTTH and THTTT

(which are all 5 long, so are all equally likely)

ok now try to think how to work out the likelihood of all the others!

EDIT: do the Ts look bolder than the Hs to you, or is it just my weird eyesight?

19. Nov 15, 2012

### h0llow

TT = 25% likely
TTHHH,etc.. = 3.125% likely (since there are 5(0.5^5))......X 8
HTT = 12.5%(0.5^3) X 2
HHTT = 6.25% X 4

everything adds up to 100% 0_o..what now..

the t's look fine

Last edited: Nov 15, 2012
20. Nov 15, 2012

### h0llow

is the answer by any chance 81.25% and 18.75% splits in prize?

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