## BdMO National Higher Secondary 2020 P1

লাজিম দুটো \(24\) তল বিশিষ্ট ছক্কা চালে। সে দুটো চালের মধ্যে যেই সংখ্যাটা বড়, সেটা নেয়। \(N\) একটা পূর্ণসংখ্যা যা \(24\)-এর চেয়ে বড় নয়। \(N\)-এর মান সর্বোচ্চ কত হতে পারে যেন বলা যাবে যে লাজিমের নেওয়া সংখ্যাটা কমপক্ষে \(N\) হওয়ার সম্ভাবনা \(>50\%\)?

Lazim rolls two \(24\)-sided dice. From the two rolls, Lazim selects the die with the highest number. \(N\) is an integer not greater than \(24\). What is the largest possible value for \(N\) such that there is a more than \(50\%\) chance that the die Lazim selects is larger than or equal to \(N\)?

Lazim rolls two \(24\)-sided dice. From the two rolls, Lazim selects the die with the highest number. \(N\) is an integer not greater than \(24\). What is the largest possible value for \(N\) such that there is a more than \(50\%\) chance that the die Lazim selects is larger than or equal to \(N\)?

This section is intentionally left blank.

- Mehrab4226
**Posts:**230**Joined:**Sat Jan 11, 2020 1:38 pm**Location:**Dhaka, Bangladesh

### Re: BdMO National Higher Secondary 2020 P1

Noob level solution

Last edited by Mehrab4226 on Tue Feb 09, 2021 7:05 pm, edited 1 time in total.

*The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.*

-Henri Poincaré

- Anindya Biswas
**Posts:**264**Joined:**Fri Oct 02, 2020 8:51 pm**Location:**Magura, Bangladesh-
**Contact:**

### Re: BdMO National Higher Secondary 2020 P1

I have another solution,

Now it should be clear that $Pr\left(max(i, j)\geq N\right)=1-\frac{(N-1)^2}{24^2}$

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."

—

—

**John von Neumann**- Mehrab4226
**Posts:**230**Joined:**Sat Jan 11, 2020 1:38 pm**Location:**Dhaka, Bangladesh

### Re: BdMO National Higher Secondary 2020 P1

The grid is actually really great(It simplifies almost everything). How did you come up with the idea?

*The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.*

-Henri Poincaré

- Anindya Biswas
**Posts:**264**Joined:**Fri Oct 02, 2020 8:51 pm**Location:**Magura, Bangladesh-
**Contact:**

### Re: BdMO National Higher Secondary 2020 P1

I actually got this as a suggestion to try to represent things in a table in probability puzzles rather than thinking about a freaking huge tree. This type of questions like "probability of sum of two numbers on two dice is equal to 8" or this particular one we are discussing are nice to visualize with a table instead of trees. If you see the probability puzzles in brilliant.org, they also used grids frequently.Mehrab4226 wrote: ↑Tue Feb 09, 2021 7:11 pmThe grid is actually really great(It simplifies almost everything). How did you come up with the idea?

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."

—

—

**John von Neumann**- Shon4poth-সঞ্চারপথ
**Posts:**1**Joined:**Mon Feb 08, 2021 10:52 pm

### Re: BdMO National Higher Secondary 2020 P1

What is the concept behind this statement: "Now it should be clear that $Pr\left(max(i, j)\geq N\right)=1-\frac{(N-1)^2}{24^2}$" , Why $(N-1)$ comes instead of just $N$? Since we have already provided that $N<24$

- Anindya Biswas
**Posts:**264**Joined:**Fri Oct 02, 2020 8:51 pm**Location:**Magura, Bangladesh-
**Contact:**

### Re: BdMO National Higher Secondary 2020 P1

We are provided that $N\leq24$. $Pr\left(max(i,j)\geq N\right)=1-Pr\left(max(i,j)\in\{1,2,3,\dots,N-1\}\right)=1-Pr\left(i,j\in\{1,2,3,\dots,N-1\}\right)$Shon4poth-সঞ্চারপথ wrote: ↑Sat Feb 27, 2021 1:27 amWhat is the concept behind this statement: "Now it should be clear that $Pr\left(max(i, j)\geq N\right)=1-\frac{(N-1)^2}{24^2}$" , Why $(N-1)$ comes instead of just $N$? Since we have already provided that $N<24$

That's why it is $(N-1)^2$

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."

—

—

**John von Neumann**