Mind Your Decisions

Successive Wins Interview Question

Presh Talwalkar — Mon, 07 Sep 2026 17:35:49 +0000

I came across an interesting problem called “successive wins” in Frederick Mosteller’s Fifty Challenging Probability Problems with Solutions. Variants of this puzzle have been used as interview questions at quant trading firms like Jane Street Capital.

Here is the setup:

Bob is an aspiring tennis player. He can earn a prize by winning (at least) 2 tennis sets in a row out of a three-set series. He has to play his teacher and the club champion alternately: either teacher-champion-teacher or champion-teacher-champion according to Bob’s choice. Everyone knows the club champion is a better player than the teacher. Which series should Bob choose?

As usual, watch the video for a solution.

Successive Wins Interview Question

Or keep reading.

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Answer To Successive Wins Interview Question

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

A game theory approach…

You can reach the answer even without doing any math. Just think like the problem maker and reason backwards.

Since the teacher is the worse player, it would make sense that Bob play him more than the champion. The choice of teacher-champion-teacher seems obvious… perhaps it’s a little bit too obvious.

This puzzle appears in a book of challenging probability problems, so the obvious answer should be held with suspicion. After ruling that choice, one can conclude the correct choice must be champion-teacher-champion.

(I know this sounds like that Princess Bride scene, but thankfully this reasoning has an ending. And in fact, champion-teacher-champion is the right answer.)

While we have reasoned to the correct answer, we have not solved a more interesting question: why should playing the tougher opponent more make Bob’s winning streak more likely?

The explanation

The reason has to do with the order of play. While champion-teacher-champion will have fewer expected wins, Bob doesn’t care about wins per se. He cares about creating a streak.

In this puzzle, Bob needs to win at least two sets in a row. How many ways is that possible? Begin by writing out the possible outcomes of the sets and identifying the win streaks. We’ll abbreviate for win (W) and loss (L):

WWW – prize
WWL – prize
WLW
LWW – prize
WLL
LWL
LLW
LLL

Notice that Bob has precisely three ways to make a streak: by winning all the sets (WWW), winning the first two (WWL), and winning the last two (LWW).

The key feature here is that Bob needs to win the second set in order to make a streak.

As the second set is critical, Bob would be wise to play the worse player–his teacher–in that slot. And that is the reason why champion-teacher-champion is the correct answer.

The size of the advantage

If we make a few more assumptions we can also calculate the size of the advantage.

Suppose Bob’s odds of winning a set depend purely on whom he plays, and suppose each set can be considered an independent event. Let t and c denote the probability Bob can beat his teacher and the champion in any given set, respectively.

Assuming independence of sets, we can then add up the probabilities of the winning events and compare their size:

Teacher-champion-teacher
WWW = tct
WWL = tc(1 – t)
LWW = (1 – t)ct
Total = ct(2 – t)

champion-teacher-champion
WWW = ctc
WWL = ct(1 – c)
LWW = (1 – c)tc
Total = ct(2 – c)

Since the club champion is a better player, c is less than t and 2 – c is larger than 2 – t. Thus ct(2 – c) is the larger quantity and it is better to play the champion twice.

Taking the example of t=0.8 and c=0.4, the chances are 51.2 percent for champion-teacher-champion and only 38.4 percent for teacher-champion-teacher. Choosing the right order increases the chances dramatically.

Special thanks this month to:

Kyle
Daniel Lewis
Lee Redden

Thanks to all supporters on Patreon and YouTube!

References

Old post
https://mindyourdecisions.com/blog/2008/12/09/a-math-problem-that-might-help-you-win-one-million-dollars/
Medium
https://medium.com/@rishidarkdevil/the-successive-wins-problem-f4d4397853d5
Fifty Challenging Problems in Probability with Solutions, By Frederick Mosteller
https://store.doverpublications.com/products/9780486134963?srsltid=AfmBOorI4gqlaBoO8CLEGhez4Wt5eqfmNi-CHlspPW6gZpEIauDA3YLN
Jane Street Capital interview question
https://www.glassdoor.com/Interview/suppose-you-player-A-player-B-are-in-a-tennis-tournament-Player-A-has-a-better-chance-of-beating-you-than-player-B-does-QTN_124563.htm

Which Curve Is Longer Green Or Blue?

Presh Talwalkar — Sun, 30 Aug 2026 22:07:20 +0000

This is adapted from questions asked in American undergraduate and graduate standardized admissions tests.

Semi-circles are constructed on AB and BC and drawn in blue, and a semicircle is constructed on AC and drawn in green.

Which curve is longer: blue or the green?

I. Blue
II. Green
III. Equal
IV. Cannot Be Determined

Many American students found this question to be difficult. Can you figure it out?

As usual, watch the video for a solution.

Which Curve Is Longer Green Or Blue?

Or keep reading.

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Answer To Which Curve Is Longer Green Or Blue?

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

A circle’s with diameter d has a circumference πd, so a semi-circle has half that length, or 0.5πd.

The blue curve has a total length of:

0.5π|AB| + 0.5π|BC|
= 0.5π(|AB| + |BC|)
= 0.5π(|AC|)

The green curve’s length is directly calculated as:

0.5π(|AC|)

Thus the two lengths are equal! This is true even if the point B is placed anywhere else along AC, even at the extreme points of A or C.

It is interesting the lengths of the two curves are always in balance no matter where the point B is placed, as the diagram bears a resemblance to the yin and yang symbol that signifies balance.

References

Previous video
https://www.youtube.com/watch?v=JNId5NmvvHg

SAT (undergraduate admissions test)
https://www.businessinsider.com/hard-sat-math-questions-test-practice-2018-3

GRE (graduate admissions test)
https://www.prepscholar.com/gre/blog/gre-geometry-review-practice/

How Did 93% Get This Wrong? Book Costs 1 Dollar Plus Half Its Price

Presh Talwalkar — Wed, 26 Aug 2026 22:08:20 +0000

About 2 years ago I posted this puzzle. The problem is going viral again, so I have posted another video. The calculations are the same but I have expanded on the history of the problem, and also included the viral poll from January 2022 showing 93% of people gave the wrong answer.

The puzzle is actually problem 37 from the famous 1956 book The Moscow Puzzles.

“A book costs $1 plus half its price. How much does it cost?”

It is a simple puzzle, but it has confused many people on the internet. As usual, watch the video for a solution.

A Classic Puzzle Baffling The Internet

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Answer To Price Of A Book Puzzle

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Suppose the book costs b dollars. Then 1 dollar plus half the price is 1 + b/2. Thus we have:

b = 1 + b/2
b/2 = 1
b = 2

The book costs $2.

I did a poll on the MindYourDecisions YouTube channel with the options $1, $1.50, $2, and “the value cannot be determined”. The poll received over 40,000 votes. About 50% agreed with the answer of $2. About 34% said there was not enough information since cost and price can be different. The remaining responses were either $1 or $1.50.

I do think a valid criticism of the phrasing is that cost and price are two different things. Cost is the expense to produce an item while price is the amount a customer pays.

But in this sentence, it should be clear that “cost” and “price” are used synonymously as is often the case in daily life. We say “a movie ticket costs $20” to refer to the price of $20 we have to pay.

Sources

Reddit today
https://www.reddit.com/r/theydidthemath/comments/1vsgrk8/request_i_need_a_second_opinion_on_this_math/
Reddit 5 years ago
https://www.reddit.com/r/theydidthemath/comments/s9k2eb/request_can_someone_do_the_math_i_swear_its_150/

The Moscow Puzzles (problem 37) by Boris A. Kordemsky, edited by Martin Gardner
http://www.math.cmu.edu/~bkell/21110-2010s/algebra-puzzles.pdf
https://store.doverpublications.com/products/9780486270784

On Fyrock’s poll, only 7% gave the answer of $2.
https://www.youtube.com/post/UgkxztB7AMWYjz7rup7eD3-22mVDyYECuVhk

On my community page, about 60% of people give the answer of $2.
MindYourDecisions poll 5 years ago
https://www.youtube.com/post/Ugzv_XryKGRUCCNDluN4AaABCQ
MindYourDecisions poll 2 years ago
https://www.youtube.com/channel/UCHnj59g7jezwTy5GeL8EA_g/community?lb=Ugkx_heZWpPNenkmphzc4IqjSnyU59PCGAeE
MindYourDecisions poll current
https://www.youtube.com/post/UgkxhM0HA5my2PYE7RCDX_1owQIdaoo7oUFZ

My post from 2 years ago
https://mindyourdecisions.com/blog/2024/04/10/price-of-a-book-puzzle-again/
https://youtu.be/0d01sKl0wtY

Singapore model
https://journals.sagepub.com/doi/10.1177/27527263221107526

Cost
https://www.merriam-webster.com/simple/cost