[wolfgang]

John and Bill wanted to play only one game Ping-pong.The first player to reach 21 points wins the game.

John is 20% better than Bill.

Bill said to John "well, you`ll win if we score as usual,cause you play better than me".

"how do you want us to score then?" asked John.

Bill:"any point I`ll score will be counted as(1.5) points to me,and for you a point will be as it is only one point,do you agree?"

" OK" said john,"but I`ll start the first serve".

Each player serves three points in a row and then switch server.

Now, who won that game?

[bushindo]

John and Bill wanted to play only one game Ping-pong.The first player to reach 21 points wins the game.

John is 20% better than Bill.

Bill said to John "well, you`ll win if we score as usual,cause you play better than me".

"how do you want us to score then?" asked John.

Bill:"any point I`ll score will be counted as(1.5) points to me,and for you a point will be as it is only one point,do you agree?"

" OK" said john,"but I`ll start the first serve".

Each player serves three points in a row and then switch server.

Now, who won that game?

Some clarifications please:

1) Does the fact that John is 20% better than Bill means that John is 20% more likely to win a point (e.g., John would on average win 6 out of every 11 points) ?

2) Does serving change the odd of a person winning the point? If so, how does that change?

Edited November 23, 2010 by bushindo

[wolfgang]

Some clarifications please:

1) Does the fact that John is 20% better than Bill means that John is 20% more likely to win a point (e.g., John would on average win 6 out of every 11 points) ?

2) Does serving change the odd of a person winning the point? If so, how does that change?

1- yes the 20% means that John has 20% more chance to score points than Bill.

2-serving will determine the sequence only.

[Guest]

Bil Wins

[wolfgang]

Bil Wins

please explain why?

[bushindo]

John and Bill wanted to play only one game Ping-pong.The first player to reach 21 points wins the game.

John is 20% better than Bill.

Bill said to John "well, you`ll win if we score as usual,cause you play better than me".

"how do you want us to score then?" asked John.

Bill:"any point I`ll score will be counted as(1.5) points to me,and for you a point will be as it is only one point,do you agree?"

" OK" said john,"but I`ll start the first serve".

Each player serves three points in a row and then switch server.

Now, who won that game?

A recursive code shows that

John wins with probability .252

[Guest]

John and Bill wanted to play only one game Ping-pong.The first player to reach 21 points wins the game.

John is 20% better than Bill.

Bill said to John "well, you`ll win if we score as usual,cause you play better than me".

"how do you want us to score then?" asked John.

Bill:"any point I`ll score will be counted as(1.5) points to me,and for you a point will be as it is only one point,do you agree?"

" OK" said john,"but I`ll start the first serve".

Each player serves three points in a row and then switch server.

Now, who won that game?

I have not computed it yet, but the answer should be a probability of Bill or John winning !

[Guest]

Seems simple enough. If John is 20% better than Bill, then Bill should score 16.8 points for every 21 points John scores. Based on their new scoring system, Bill get 1.5 points every time he scores...so he only needs to score 14 times to have 21 points. Knowing this, Bill will score 14 times BEFORE John scores 21 times. Bill son the game.

[Guest]

Sorry, I didn't know how to hide my answer...I am new to this. And I also meant to say that Bill Won the game.

Edited November 23, 2010 by rob_3

[Guest]

Sorry to continue to be a grouch, but this is another example where the question is so poorly written that it renders the answer meaningless. The clarification helped, but it's still impossible to know the intent. I suspect that rob_3's approach is the correct one. If so, the problem might be reworded thusly:

John and Bill are playing a single game of Ping-pong.The first player to reach 21 points wins the game.

John scores 20% more points than Bill.

Bill says to John, "Well, you`ll win if we score as usual, because you play better than me".

"How do you want us to score then?" asks John.

Bill replies, "Any point I score will be counted as 1.5 points to me, but for you a point will simply be worth only one point. Do you agree?"

"OK" says John, "But I`ll serve first".

Each player serves three times in a row and then the server changes.

Now, who will win that game?

It should be noted that, under the rules of table tennis, the person serving changes hands after two points are scored. Why they are doing something unorthodox may be the author's attempt to mislead. Or the author may be unaware of the fact that the scoring is not dependent on who served.

[bushindo]

Sorry to continue to be a grouch, but this is another example where the question is so poorly written that it renders the answer meaningless. The clarification helped, but it's still impossible to know the intent. I suspect that rob_3's approach is the correct one. If so, the problem might be reworded thusly:

John and Bill are playing a single game of Ping-pong.The first player to reach 21 points wins the game.

John scores 20% more points than Bill.

Bill says to John, "Well, you`ll win if we score as usual, because you play better than me".

"How do you want us to score then?" asks John.

Bill replies, "Any point I score will be counted as 1.5 points to me, but for you a point will simply be worth only one point. Do you agree?"

"OK" says John, "But I`ll serve first".

Each player serves three times in a row and then the server changes.

Now, who will win that game?

It should be noted that, under the rules of table tennis, the person serving changes hands after two points are scored. Why they are doing something unorthodox may be the author's attempt to mislead. Or the author may be unaware of the fact that the scoring is not dependent on who served.

I don't think that this question is "so poorly written that it renders the answer meaningless". I understand the puzzle just fine, as do The Genius of Genius, Dhanannjay Deo, rob_3, and I suspect many others. This is a very nice puzzle that demonstrates the elegance of recursive functions, and makes a valuable contribution to this Logic/Math puzzle forum.

We have been through this before. I recommend that you read these excellent responses from and to your last post.

[superprismatic]

Nice Puzzle! I don't know where you get them, but keep them coming.

Bill will win with probability .748345.

[superprismatic]

I don't think that this question is "so poorly written that it renders the answer meaningless". I understand the puzzle just fine, as do The Genius of Genius, Dhanannjay Deo, rob_3, and I suspect many others. This is a very nice puzzle that demonstrates the elegance of recursive functions, and makes a valuable contribution to this Logic/Math puzzle forum.

We have been through this before. I recommend that you read these excellent responses from and to your last post.

Recursive functions? I just noticed that! I only summed the terms of a simple series. I understand that you could do this with recursive functions, but it seems like overkill in this case.

I guess we all have our different ways of approaching things. I suppose that's why maths are so cool!

[Guest]

Since the serve makes no difference on the odds of winning, every 3 serve segment is the same. ON AVERAGE

John scores 1.2 / 2.2 *3 = 1.64 points per segment

Bill scores (3 - 1.64) * 1.5 = 2.05 points per segment

The odds of Bill winning are huge

[Guest]

I like John to win the game. Everytime he scores a point, he's awarded a point and a half, which is a 50% advantage to Bill's 20% advantage of playing better. To reach 21, John needs only 14 points to Bills 21. Sound right?

I may have reversed the names. I like the underdog! How's that?

Edited November 24, 2010 by Joey D

[bushindo]

Recursive functions? I just noticed that! I only summed the terms of a simple series. I understand that you could do this with recursive functions, but it seems like overkill in this case.

I guess we all have our different ways of approaching things. I suppose that's why maths are so cool!

I didn't think about summing a series neither. Do you mean summing the chance that John won given that Bill has 0 points, 1 points, and so on? I didn't think about it that way, although now I wonder why I didn't think of that before. Recursive functions are actually fit this problem quite well. It only takes a couple of lines as in the following pseudo-code.

%%% prob( J, B ) is a function that gives the probability that John wins given that John has J points and Bill has B points.

prob = function( John_score, Bill_score )

     if John_score == 21

          return 1


     if Bill_score == 21

          return 0


     return ( (6/11)* prob( John_score + 1, Bill_score ) + (5/11)* prob( John_score , Bill_score + 1.5 ) )

endFunction


answer = prob( 0, 0 )

And yes, math is very cool!

Edited November 24, 2010 by bushindo

[Guest]

John and Bill wanted to play only one game Ping-pong.The first player to reach 21 points wins the game.

John is 20% better than Bill.

Bill said to John "well, you`ll win if we score as usual,cause you play better than me".

"how do you want us to score then?" asked John.

Bill:"any point I`ll score will be counted as(1.5) points to me,and for you a point will be as it is only one point,do you agree?"

" OK" said john,"but I`ll start the first serve".

Each player serves three points in a row and then switch server.

Now, who won that game?

Lets analyze the two possibilities

1. Assume John has Won

Score of John = 21 (since he have won). At this point Bill should have scored 20% less than 21

20% of 21 = 4.2 so Bill should have scored 21 - 4.2 = 16.8. But for bill each point is counted as 1.5 so bill score would be 16.8 * 1.5 = 25. 2 which is the score greater than John. So Its a contradiction to our assumption and so John must not have won

2. Assume Bill have won

Score of Bill= 21 (since he have won). But for Bill 1 point = 1.5 so truly Bill have scored 14 time. Since John is 20% better he would have scored 14 + (20% of 14) = 14 + 2.8 = 16.8 and his score is 16.8 So when Bill's score is 21, johns score would be 16.8 (=17) in which case Bill must have won.

From both the above cases we have proved that Bill have won

Hope my proof is correct and I have made it clear

[Guest]

Since John wins 20% more his probability of winning a point is ~ .55 while Bill's is .45. So after two points the probabilities are: John 2-0 = .55x.55, 1-1 (which is really 1-1.5) = 2 x .55x.45 [the two is in there since there are two ways of getting to 1-1, ingeneral you use the binomial coefficients]and 0-2 (which is really 0-3) = .45x.45. You can do this for each additional point.

Edited November 24, 2010 by easygoer

[Guest]

I think it depends on what is meant by "John is 20% better than Bill"

If it means, (the likely meaning) that John wins 20% more points than Bill does, then

For every point that bill wins, he gets 1.5 points, while Jim wins 1.2 and gets 1.2

So, Bill wins

If it means that odds of John winning a point are 20% higher than Bill such that Odds of John winning a point-60%, Odds of Bill winning a point-40%, then

I am not good at math and will only use logic, so my answer may sound very convoluted, please bear with me

Since, the serve doesn't make any difference, I'm gonna approach this in sets of five to get whole points (which is what you get in a real game) rather than fractions

After 5 points, John wins 3 and Bill wins 2, which would award John 3 points and Bill 3 points as well (2 X 1.5)

After 30 serves, they are both at 18 points each. At this point, John needs to win 3 points while Bill needs to win 2 points to win the game. The question is, who is more likely to do it first?

Now we have 4 situations for the next 3 serves after 30

1. Bill wins 2 of the first 2 and wins

Odds of Bill winning (A) : 0.4 X 0.4 = 0.16

2. Bill wins 2 of three and wins

Odds of Bill winning (B) : 0.4 X 0.6 X 0.4 X 2 (2 permutations) = 0.192

3. Bill wins none and John wins

Odds of John winning this way (x): 0.6 X 0.6 X 0.6 = 0.216

4. Bill wins one and John wins two

Odds of this event: 0.6 X 0.6 X 0.4 X 3(3 permutations) = 0.432

So here at 33 serves, John is at 20 points and Bill is at 19.5

Now, the next serve will decide the game but the odds over the next one will be affected by the assumed outcome of last 3 since the last 3 serves cannot be taken as an independent event till it happens

Remember that when you consider a total of 4 serves after 30, John is likely to win 2.4 of those while Bill will win 1.6

Odds of John winning the next is 0.4 (to make it to a total 2.4 / 4 serves)

Overall Odds of John winning like this if this event were to happen (y) : 0.4 X 0.432 = 0.1728

Odds of Bill winning the next is 0.6 (to make it to a total of 1.6 / 4 serves)

Overall odds of Bill winning this way if this event were to happen ©: 0.6 X 0.432 = 0.2592

NOW THE CONCLUSION:

Odds of Bill winning = A + B + C = 0.16 + 0.192 + 0.2592 = 0.6112

Odds of John winning = x + y = 0.216 + 0.1728 = 0.3888

BILL STILL WINS

I'm sure some of you can do this in a much simpler way, Thanks for reading!

Edited November 24, 2010 by HB921

[wolfgang]

Sorry to continue to be a grouch, but this is another example where the question is so poorly written that it renders the answer meaningless. The clarification helped, but it's still impossible to know the intent. I suspect that rob_3's approach is the correct one. If so, the problem might be reworded thusly:

John and Bill are playing a single game of Ping-pong.The first player to reach 21 points wins the game.

John scores 20% more points than Bill.

Bill says to John, "Well, you`ll win if we score as usual, because you play better than me".

"How do you want us to score then?" asks John.

Bill replies, "Any point I score will be counted as 1.5 points to me, but for you a point will simply be worth only one point. Do you agree?"

"OK" says John, "But I`ll serve first".

Each player serves three times in a row and then the server changes.

Now, who will win that game?

It should be noted that, under the rules of table tennis, the person serving changes hands after two points are scored. Why they are doing something unorthodox may be the author's attempt to mislead. Or the author may be unaware of the fact that the scoring is not dependent on who served.

Thanks dear friend for the new form..I said before,I am so sorry for my bad English...

If I bother some people,,so I`ll stop writing..

moreover I know the ping-pong rules EXACTLY!!

but I wanted them to play this game according to my rules (if you don`t mind).

and I wanted them to change serve after each three points.

please tell me,wether I can write other puzzles in the future or not..

And I`d like to ask all my friends the same guestion...

Thank you all

[wolfgang]

I don't think that this question is "so poorly written that it renders the answer meaningless". I understand the puzzle just fine, as do The Genius of Genius, Dhanannjay Deo, rob_3, and I suspect many others. This is a very nice puzzle that demonstrates the elegance of recursive functions, and makes a valuable contribution to this Logic/Math puzzle forum.

We have been through this before. I recommend that you read these excellent responses from and to your last post.

Dear Bushindo! I am very happy to see you defending my situation...I like this forum and I like all people here..

but If my bad English bothers anyone...so I`ll stop writing.

Thank you again ...best regards!!

[wolfgang]

Nice Puzzle! I don't know where you get them, but keep them coming.

Bill will win with probability .748345.

Thanks...I am an old man(57)and in my head are thousands of things...

I hope I don`t bother you due to my bad English..

Edited November 24, 2010 by wolfgang

[wolfgang]

Lets analyze the two possibilities

1. Assume John has Won

Score of John = 21 (since he have won). At this point Bill should have scored 20% less than 21

20% of 21 = 4.2 so Bill should have scored 21 - 4.2 = 16.8. But for bill each point is counted as 1.5 so bill score would be 16.8 * 1.5 = 25. 2 which is the score greater than John. So Its a contradiction to our assumption and so John must not have won

2. Assume Bill have won

Score of Bill= 21 (since he have won). But for Bill 1 point = 1.5 so truly Bill have scored 14 time. Since John is 20% better he would have scored 14 + (20% of 14) = 14 + 2.8 = 16.8 and his score is 16.8 So when Bill's score is 21, johns score would be 16.8 (=17) in which case Bill must have won.

From both the above cases we have proved that Bill have won

Hope my proof is correct and I have made it clear

[wolfgang]

I think it depends on what is meant by "John is 20% better than Bill"

If it means, (the likely meaning) that John wins 20% more points than Bill does, then

For every point that bill wins, he gets 1.5 points, while Jim wins 1.2 and gets 1.2

So, Bill wins

If it means that odds of John winning a point are 20% higher than Bill such that Odds of John winning a point-60%, Odds of Bill winning a point-40%, then

I am not good at math and will only use logic, so my answer may sound very convoluted, please bear with me

Since, the serve doesn't make any difference, I'm gonna approach this in sets of five to get whole points (which is what you get in a real game) rather than fractions

After 5 points, John wins 3 and Bill wins 2, which would award John 3 points and Bill 3 points as well (2 X 1.5)

After 30 serves, they are both at 18 points each. At this point, John needs to win 3 points while Bill needs to win 2 points to win the game. The question is, who is more likely to do it first?

Now we have 4 situations for the next 3 serves after 30

1. Bill wins 2 of the first 2 and wins

Odds of Bill winning (A) : 0.4 X 0.4 = 0.16

2. Bill wins 2 of three and wins

Odds of Bill winning (B) : 0.4 X 0.6 X 0.4 X 2 (2 permutations) = 0.192

3. Bill wins none and John wins

Odds of John winning this way (x): 0.6 X 0.6 X 0.6 = 0.216

4. Bill wins one and John wins two

Odds of this event: 0.6 X 0.6 X 0.4 X 3(3 permutations) = 0.432

So here at 33 serves, John is at 20 points and Bill is at 19.5

Now, the next serve will decide the game but the odds over the next one will be affected by the assumed outcome of last 3 since the last 3 serves cannot be taken as an independent event till it happens

Remember that when you consider a total of 4 serves after 30, John is likely to win 2.4 of those while Bill will win 1.6

Odds of John winning the next is 0.4 (to make it to a total 2.4 / 4 serves)

Overall Odds of John winning like this if this event were to happen (y) : 0.4 X 0.432 = 0.1728

Odds of Bill winning the next is 0.6 (to make it to a total of 1.6 / 4 serves)

Overall odds of Bill winning this way if this event were to happen ©: 0.6 X 0.432 = 0.2592

NOW THE CONCLUSION:

Odds of Bill winning = A + B + C = 0.16 + 0.192 + 0.2592 = 0.6112

Odds of John winning = x + y = 0.216 + 0.1728 = 0.3888

BILL STILL WINS

I'm sure some of you can do this in a much simpler way, Thanks for reading!

Thats ok..I am quite satisfied!

[superprismatic]

Thanks...I am an old man(57)and in my head are thousands of things...

I hope I don`t bother you due to my bad English..

Bother me? No! On the contrary, I enjoy your puzzles very much.

If I am unsure about some parts of a puzzle, I'll do what Bushindo

did and simply ask about it. I don't understand why anyone would

do otherwise. Please keep your puzzles coming.

[superprismatic]

I didn't think about summing a series neither. Do you mean summing the chance that John won given that Bill has 0 points, 1 points, and so on? I didn't think about it that way, although now I wonder why I didn't think of that before. Recursive functions are actually fit this problem quite well. It only takes a couple of lines as in the following pseudo-code.

%%% prob( J, B ) is a function that gives the probability that John wins given that John has J points and Bill has B points.
prob = function( John_score, Bill_score )
if John_score == 21
return 1

if Bill_score == 21
return 0

return ( (6/11)* prob( John_score + 1, Bill_score ) + (5/11)* prob( John_score , Bill_score + 1.5 ) )
endFunction

answer = prob( 0, 0 )

And yes, math is very cool!

I didn't worry about the 1.5 points part for Bill,

I just converted it into the equivalent problem

where each gets 1 point from winning a rally but

Bill wins if he gets 14 points before John gets 21.

So, the probability that Bill wins is just

(5/11)¹⁴ × Σ _(13+i)C_i × (6/11)ⁱ

where the summation is from i=0 to 20.

Your recursive function approach is nice because you

don't have to concern yourself with the nitty-gritty

of the binomial coefficients and such -- it all comes

out in the recursive wash. I rarely think of taking

this approach because the recursive stack sucks up

memory very quickly. This was a real problem with

this approach in the days when memory was small.

Even now, one needs to be mindful of this fact when

the recursion bushes out a lot.

Edited November 24, 2010 by superprismatic