Winning a bet

Probability Level 1

Trevor and I were playing golf.

As we all know, Trevor is a pro golfer and I don't know squat about golf (except the fact that we are supposed to hit golf balls with a club).

Now, he made a bet with me. He said, "We will play $N$ matches where you're gonna choose the value of $N$ . If you win at least one match, I'll pay you $\$100$ since I'm super rich and if you don't, you'll pay me $\$1$ "

Now, I opened my wallet and I saw that it was emptier than a banker's heart. Amongst the cobwebs and flies in it, luckily, there was a $\$1$ bill. Naturally, I accepted the bet since I'm almost broke and need cash.

Generally, if I were to play a match against him, I would have a probability of $0$ of winning. But since here's money involved, the probability of me winning a single match against him is boosted to $\dfrac{1}{3}$ .

I want to have at least a $90\%$ chance of winning the bet. What is the minimum value of $N$ that I should choose?

The answer is 6.

6 solutions

Trevor Arashiro
Mar 25, 2015

Correct answer: 0, cuz I'll always win :3

Haha. If I may ask, what is your 'cap? I wonder if there is another question that can come out of Prasun's, like, "If Trevor and Matthew have 'caps of 5 and 10, respectively, what is the probability that Matthew will beat Trevor straight up?"

P.S.. I wrote a short treatise on AI here , in case you're interested in carrying on that discussion. :)

Brian Charlesworth - 6 years, 2 months ago

My handicap is 1.4 and dropping currently, up from 0.5 two months ago.

That does seem like a simple question at first thought. But when I think about it, that's a very very complicated question.

Here is a possible phrasing. (Ignore the lower case letters for now, I explain them below)

After a set of 5 (p) matches, Trevor and Prasun have average scores of 75 (a) and 80 (b), respectively. Trevor's scores range between 68 (x) and 80 (y) inclusive (note that this does not necessarily mean that 68 and 80 are two of his scores). Prasun's scores range between 71 (t) and 86 (s) inclusive.

What is the probability that Prasun wins at least 2 (n) of the 5 matches?

The extreme possibilities are that Prasun and I shoot our average score after every match, in which case I will win every time. However, say that my scores are 80,80,75,70,70 and Prasun's are 79,79,74,84,84. He will win 3 of the matches.

Unfortunately, in a combinatorics or death situation, I'd chose the latter. I don't know how to solve this without the use of excessive bashing or a computer program, although I know that this is solvable..

Remember the (x)'s in the previous question, those denote a variable that can be manipulated. This problem is very versatile.

Now for the fun part where those variables look a little more menacing.

There are two players, player $A$ with data set $A$ and elements $a_i$ and player $B$ with data set $B$ and elements $b_i$ . The arithmetic mean of each data set is 10 and 20, respectively.

A "game" is played in which each $a_i$ is compared to its corresponding $b_i$ with the same base. If the element from set $A$ is less than its corresponding element from set $B$ , then player $A$ is awarded 1 point and visa versa with player $B$ .

What is the probability that player $B$ has more points than player $A$

Also, is a fixed number of matches needed for this?

I'll post the two examples in a note.

Btw, I just read your "AI" comment from that problem, and it's a lot to digest. I'll leave a full response soon.

Trevor Arashiro - 6 years, 2 months ago

Holy smokes! Do you have any thoughts of going pro in the future? Sounds like there will be both academic and sports scholarships headed your way. :)

Yeah, golf lends itself to statistical analysis, and apparently you've already given it a lot of thought. I did realize that the question I posed was deceptively complicated; it would depend on the means and spread of the two players' distributions of scores, so it would probably be more easily dealt with using statistics. However, with the "data sets" you mention, one thought would be to give a list of, say, of ten scores for each player obtained in 10 head-to-head stroke play matches, but give them in random order. The question would then be to determine the expected number of matches won by each player, possibly after factoring in handicaps. I suppose this is just a variation of your proposed problem. A relatively small fixed number, (no more than 10), would probably be necessary to make the problem tractable; any more than that and we'd need to get into $\mu$ and $\sigma$ calculations.

P.S.. No rush on the AI thing; I took such a long time to respond that I wasn't sure if you had noticed, that's all. :)

Brian Charlesworth - 6 years, 2 months ago

@Brian Charlesworth – Haha, thank you. But I'm not looking at turning professional. I'd rather become an investor or CFO. I'm looking to get an MBA and minor in comp sci and software engineering. What are your degrees in and from what college?

Btw, I don't even know what those symbols $\mu$ $\sigma$ even mean. Lol

Trevor Arashiro - 6 years, 2 months ago

@Trevor Arashiro – Oh, those are just for the mean and the standard deviation of a distribution. lol

Wow, those are wise goals you have, and extremely marketable. Pro golf can be lucrative, but only a few really make a go of it and I'm not sure how fulfilling a life choice it would be. Being good at golf, though, is a useful part of ones skill set in the business world; a lot of deals and negotiations are done over a round of golf. :) I was far less practical than you at your age; I did a double honours in math and physics at my local university. :P

Brian Charlesworth - 6 years, 2 months ago

@Brian Charlesworth – Well, That explains why you're so unbelievably good at math. ;)

But I've also noticed that your english is out standing. Your articulate and sophisticated diction are absolutely amazing. While reading your brilliant comment on AI, I was perplexed by the many sentences that I can only describe as... Well, it's hard to put into words, but maybe, "beautiful use of the english language"?. Is this just a skill you picked up along the way?

Trevor Arashiro - 6 years, 2 months ago

@Trevor Arashiro – That's the nicest compliment I've had in a while. Thank you. I do enjoy mathematics, but my favorite topics of conversation are history, politics and philosophy, so I suppose that any writing 'style' I might have comes via osmosis from all the books I've read on these subjects. In its own way the study of mathematics helps as well, in that it shares the same ideals as writing, namely clarity, elegance and efficiency.

Brian Charlesworth - 6 years, 2 months ago

I read that as "What is your crap?" :P

Prasun Biswas - 6 years, 2 months ago

Haha. "'cap" is short for "golf handicap", which is, roughly, the average number of shots over par you can expect to shoot over 18 holes. Your level of sarcasm is impressive, Prasun. :P

Brian Charlesworth - 6 years, 2 months ago

@Brian Charlesworth – Thanks for the compliment. I was going to make another joke about "holes" but then I changed my mind. :3

Prasun Biswas - 6 years, 2 months ago

@Prasun Biswas – Good decision. :)

Brian Charlesworth - 6 years, 2 months ago

@Brian Charlesworth – Yes, indeed. :3

Prasun Biswas - 6 years, 2 months ago

@Brian Charlesworth – I've actually never heard that term used before. Lol, maybe it's more of a mainland thing

Trevor Arashiro - 6 years, 2 months ago

@Trevor Arashiro – I get the feeling that "it's more of a mainland thing" is a common phrase in the Aloha State. :) Maybe it's more of an age thing; I haven't been an avid golfer, or even a casual one, for decades now, (it was going to become prohibitively expensive to become a full member), so the lingo may have changed. I also was starting to find the game to be "a good walk spoiled", as Mark Twain once stated. I still like watching the majors though, especially The Masters.

Brian Charlesworth - 6 years, 2 months ago

Upvoted! :3

Prasun Biswas - 6 years, 2 months ago

:3 $\color\white{. }$

Trevor Arashiro - 6 years, 2 months ago

A Former Brilliant Member
Mar 25, 2015

The probability of Trevor winning a match is 2/3. The chance of Parun to win a match can only be atleast 90 percent if the chance of Trevor winning all the match is less than or equal to 10 percent. Let there be N matches than probability of Trevor to win all matches is $(2/3)^{N}$ which should be less than or equal to 0.1.

For this to happen the minimum integer value of N is 6 & the answer is 6

You should show how you solved the inequality $\displaystyle\left(\dfrac{2}{3}\right)^N\leq 0.1$

Hint: Apply logarithm on both sides of the inequality to be solved making use of the fact that,

$0\lt a\leq b\implies \log_x a\leq \log_x b~,~x\in (1,+\infty)$

Prasun Biswas - 6 years, 2 months ago

Oh! The inequality! I haven't shown it cause I am still learning LATEX and cannot use different symbols. Can you suggest some source from where I can learn it.

A Former Brilliant Member - 6 years, 2 months ago

These links might be helpful.

Beginner's $\LaTeX$ guide

Free $\LaTeX$ editor

A note providing links to $\LaTeX$ learning sources (e-books)

Prasun Biswas - 6 years, 2 months ago

@Prasun Biswas – Thanks! I will try to learn as fast as I can.

A Former Brilliant Member - 6 years, 2 months ago

Brock Brown
Mar 25, 2015

Python 2.7:

from random import randint
def prob_of_win(N):
    trials = 100000
    wins = 0.0
    for trial in xrange(trials):
        for game in xrange(N):
            # if I won the game
            if randint(1,3) == 1:
                wins += 1
                break
    return wins/trials
N = 1
while prob_of_win(N) < 0.9:
    N += 1
print "Answer:", N

Boy, you and your ever useful comp sci skills :). It seems that you really love working with Python code on all types of problems and you have an extensive knowledge of how to use it.

Trevor Arashiro - 6 years, 2 months ago

Thanks. :D

I hope you don't mind if I put that quote on my resume. Jimmy John's bicycle delivery ain't going to cut it forever.

Brock Brown - 6 years, 2 months ago

I would be honored if you did so.

What job are you applying for? Are you going to be working programming with some of the tech giants? Judging by your age, you are probably in college right now. Which university are you attending right now and what are you planning to major in?

Trevor Arashiro - 6 years, 2 months ago

@Trevor Arashiro – I'm not sure what kind of work I can apply for yet. I have a general idea of how to script but sometimes I feel like what I know has almost zero practicality in most real programming jobs. I'd love to work for Google, but everyone and their mom wants to work for Google.

No college for me right now; I learn mostly from edX , MIT OpenCourseWare , and Wikipedia . I made a dumb decision when I was in high school that I didn't need college to learn and I thought that this is the age where you could learn everything you need to learn from the internet. Big mistake. It is incredibly hard to get started without college. Since my GPA is down the crapper I need to study my butt off for the ACT if I want to go.

What about you? What are you planning on doing with your life?

Brock Brown - 6 years, 2 months ago

@Brock Brown – Mm, don't worry, we all make bad decisions once in a while. But luckily you can recover from this one. I'd recommend going to any college for now. Then once you show your true potential as a student, you'll without a doubt get accepted into a better college. Transferring colleges has become more common (or at least it seems) as every college I visited on my trip talked about transfer students during admissions.

I am looking to get an MBA and minor in comp sci and software engineering. I hope to work as a CFO or stock broker on Wall Street one day. Lol, it probably won't happen, but I'll try to get there the best I can.

Trevor Arashiro - 6 years, 2 months ago

Ferran Espuña Bertomeu
Mar 31, 2015

Probability of not winning a match = 2/3

Probability of winning N matches = 1-(2/3)^N

Probability of winning N matches ≥ 9/10

1-(2/3)^N≥9/10 » (2/3)^N≥1/10 »N≥log(2/3)1/10 » N≥5.678

Minimum whole value of N is 6

Bostang Palaguna
Aug 18, 2020

$1 - (\frac{2}{3})^n > \frac{9}{10}$

the minimum $n$ is 6

Pete Boulton
Mar 30, 2015

0.66*6<0.1

$0.66*6=3.96\not\lt 0.1$

Prasun Biswas - 6 years, 2 months ago

Winning a bet

The answer is 6.

6 solutions

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