# Homework Help: Percentage and probability problem

1. Mar 19, 2013

### reenmachine

1. The problem statement, all variables and given/known data

Suppose I play a game and have a 8,33% chance to lose and a 5,7% chance to win (with the additionnal 85,97% forcing me to play again until I win or lose) does this mean I have a 40,6% chance to win?

2. Relevant equations

8.33 + 5.7 = 14.03
8.33/14.03*100 = 59,4% chance to lose
5.7/14.03*100 = 40,6% chance to win

Is this the correct way to calculate it?

thanks

Last edited: Mar 19, 2013
2. Mar 19, 2013

### Staff: Mentor

No, just like it says in the first sentence above, you have an 5.7% chance of winning and an 8.33% chance of losing.

3. Mar 19, 2013

### reenmachine

I probably wasn't clear enough , but if you miss both the winning and losing ''points'' or whatever you want to call it , then you have to play again (with the same odds) until you win or lose , so the sum of the percentage of chances of losing and winning have to be 100% no?

There's a 100% chance that the outcome will be winning or losing if you play until you win or lose , unless you always hit that 85,17% until infinity , is that what you were referring to?

Are my calculations correct from that perspective?

4. Mar 19, 2013

### Staff: Mentor

I don't think it is. What I would do is make a table of the outcomes for 1 roll, 2 rolls, 3 rolls, 4 rolls, and so on (I'm assuming that each game involves the roll of a die, but whatever).

For the first game/roll, the probabilities are 5.7% (win), 8.33% (lose), 85.17% (continue) - BTW, those don't add up to 100%. Should that last number be 85.97%?Check your problem statement.

For the second and subsequent games/rolls, it must have been that the first roll was neither a win nor a loss, so conditional probabilities need to be calculated.

5. Mar 19, 2013

### reenmachine

sorry for the 85,17% just a brain cramp.

I'm not sure I understand you though , if I have to calculate my chance of winning in this game I can't just go on forever about the 1st , 2nd , 3rd , 4th ... infinity rolls and so on.I want to calculate my odds of winning versus losing from before the game , so how do I make those conditionnal probabilities you are talking about?

I'm lost. :(

6. Mar 19, 2013

### Staff: Mentor

7. Mar 19, 2013

### reenmachine

oh well , I guess that's over my head for the time being.Trying to understand this wikipedia page but it's pretty complicated.

thanks anyway!

8. Mar 19, 2013

### Ray Vickson

Yes, you are correct, but you do not give a good "explanation". What you are calculating is the conditional probability P{win the game} = P{win the round|(win the round) or (lose the round)}.

Another way to see this is: P{win} = P{win on round 1} + P{win on round 2} + P{win on round 3} + ... (an infinite series). You win on round 1 with probability 0.057, you win on round 2 if you "continue" from round 1 and then win round 2, etc. Can you calculate those probabilities? If so, just add them all up.

9. Mar 19, 2013

### reenmachine

I'm not sure I understand everything in your post Ray , but it is getting clearer.

By ''P{win the game} = P{win the round|(win the round) or (lose the round)}'' you mean the probability of winning the game is equal to the probability of winning the round or having to win or lose the next round (basically hitting the 85.97% chance of continuing) and so on?

If I have a 0.057% chance of winning round 1 , then I have a 0.8597(0.057) chance of winning in round 2 , then a 0.8597(0.8597)(0.049) chance of winning in round 3 and so on? Am I completely off the track here? Or is it 0.8597(0.8597)(0.057) for round 3? Or is my 0.8597(0.8597) completely irrelevant?

( 0.049 comes from 0.8597(0.057) )

btw can't thank you guys enough for the help you provide even if I feel like a dumbass sometimes. ;)

10. Mar 19, 2013

### Staff: Mentor

This is what I was alluding to in my hints...

11. Mar 19, 2013

### reenmachine

although it is clear that I don't understand completely , how can you add up an infinite series if I want to know the odds of winning the game BEFORE starting round 1?

What I mean is , when will I know when to stop adding up , or at which round to stop? (even if I knew how to calculate the conditionnal probabilities , which I don't it seems)

So how does my 0.8597(0.057) for round 2 probability of winning sounds?

Take note that this isn't a homework , but a personal curiosity , I just decided to write it here as the stuff in the other subforums are quite advanced and I didn't want to bore people out of my thread :X (why I'm saying this is I'm not currently taking a probability class or any class for that matter)

thanks !!!!

Last edited: Mar 19, 2013
12. Mar 19, 2013

### reenmachine

oh and btw ray , you said I was correct (even though my explanation wasn't) , did you mean my odds were indeed 40,6% to win the game overall before round 1 but that I didn't make the right calculations to come to this conclusion

13. Mar 19, 2013

### Ray Vickson

Your calculation was OK, but an *explanation* of how you obtained it is lacking; that is, why do you calculate 8.33 + 5.7 = 14.03 and then divide by it? Also, you rounded off the answer from 0.4062722737 to 0.406; this is OK, but you should say that you did that (just so the person marking you work knows you understand).

14. Mar 19, 2013

### reenmachine

as I said in my previous post Ray , this isn't a homework , I cheated a bit on which subforums to post it.I have no ''explanation'' to give to anybody per say , nobody is going to mark that work , I only want to understand how to calculate the odds of winning this game for my personal curiosity :X

are you saying 40,6% is the right answer to the question: what are the odds of winning this game?

so are the conditionnal probabilities basically useless if I really want to quickly obtain my answer for my own self when I can proceed as I did in the OP without using them at all?

And as for the explanation , if I really had to explain it , I guess I would say something like: I intuitively added 8.33 and 5.7 (for 14.03) so that I only count the odds of winning or losing the game while ignoring the rest of the odds which are only taking me back to the same situation in the next rounds to infinity.By doing this , I can make a fraction with each percentage (winning(5.7) or losing(8.33)) out of the overall relevant odds that the game will end this round (14.03).So I have a 5.7 out of 14.03 chance of winning the game and a 8.33 out of 14.03 chance of losing the game.A fraction is a division so I divided and it gave me 0.406... for winning , and I multiply by 100 in order to make it as a percentage , which gave me 40.6%.

cheers!

Last edited: Mar 19, 2013
15. Mar 20, 2013

### Ray Vickson

First: conditional probabilities are not "useless"; they are crucial to getting the correct answer in this problem. The answer IS a conditional probability!

Second: forget about the terminology "odds". In this problem, you are talking about *probabilities*. You might be choosing to use the word 'odds' because it is shorter than 'probabilities', but make sure you really know the difference. They are not the same!

Third: whether or not anybody will be marking this, it still is a good idea to deal properly with floating-point numbers. Never, never, never round off prematurely; doing so can (in some problems a bit more complicated than this one) come back to bite you: you can get sometimes get seriously incorrect results if you do not carry enough significant figures throughout a calculation. When writing the answer, then sure, round off at that point. (Note, however, that if you were planning to use the answer to the current question in some additional calculations, then rounding off 0.4062722737 to 0.406 might produce undesirable errors later.)

I am not saying that what you did was wrong; I am just saying that you should *understand* what you did and why you did it.

16. Mar 20, 2013

### reenmachine

Oh when I do calculations I normally don't round off at all , I just did here for the sake of being clear on the forum , which was apparently a mistake.It's funny that you tell me that now , I still remember back in the days in highschool when my math teachers used to ask me to round off where I would literally write numbers such as 42.8738738738 instead of 43 or 42.9.

So let.s suppose I'm right about the 40,6% , is this the odds or the probability of winning?

As far as why I proceeded with the 5.7 + 8.33 = 14.03
5.7/14.03*100 it's true that I simply used my intuition.

I would still like to know clearly if that's what I'm looking for because I am now confused.You are not telling me that I'm wrong , but you're not telling me that I'm right either , which is more confusing than anything :X

cheers

Last edited: Mar 20, 2013
17. Mar 20, 2013

### Ray Vickson

No, rounding is not a mistake, but it is advisable to show that you *know* about it, at least if you were to hand in work for someone else to mark.

You did nothing wrong: I just could not tell from your submission whether or not you were aware of certain issues, so I wanted to draw your attention to them. Since you say you are aware of them, fine---just ignore what I said!

However: I really do insist that your use of "intuition" is generally a bad idea when doing probability calculations. You happened to get the correct answer in this case, but often such an approach can lead to serious errors. Probability is funny that way: I have heard it said that more professional mathematicians have made blunders in probability theory than in any other subject. That is the reason I argue against intuition---until your intuition has become "trained" by doing lots of examples systematically. In particular, you really do need to become familiar with the concept of conditional probability and its uses; it is not that difficult, especially if you do a Google search for on-line articles on the topic.

18. Mar 20, 2013

### reenmachine

I'm not sure if I'm aware of certain issues , I would need to know which issues you are talking about :X

If you're talking about the rounding off issue , then I'm fine with the why , when and how to round off.

I guess the ''conditionnal probability'' concept came to me intuitively , but I was unaware of the formulas and technics to use them.I understand that intuition isn't that good , especially for probabilities , but I had to work with what I had.Like I said , nobody is going to mark that work and the answer to the specific problem I'm looking for isn't going to come to me by some textbook or teacher.I won't ever know the answer unless I'm completely sure of my math , which is why I asked for help here.

cheers!

19. Mar 20, 2013

### Ray Vickson

OK, you were working with what you had. I am urging you to go beyond what you have, and "get more".

Those are my last words on this topic.

20. Mar 20, 2013

### reenmachine

absolutely!!! Completely agree that I need to get more , and I will :)

cheers and thanks for your help Ray!!!