BdMO National Higher Secondary 2020 P1

[Mursalin]

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 \(N\)?

[Mehrab4226]

Noob level solution

N = 17
Let the two dice be, $D_1$ and $D_2$
Now, If $D_1$ puts out $17$ and $D_2$ can put out any of ${1-16}$ So there are $16$ such outcomes where $D_1$ has the highest value and it gives us something $\geq N$.
Similarly, if $D_1$ puts out $18$ then there are $17$ such outcomes and so on.

Thus Lazim takes a number greater than or equal 17 when $D_1$ has the highest in $(16+ 17+ \cdots 22+23)$
Thus the same thing can be said of $D_2$ had the highest value. And again we excluded the outcome when both die have the same value there are 8 such outcomes.

Thus the total number of outcomes which are favourable $(16+ 17+ \cdots 22+23)\times 2 + 8= 320$


Total number of outcomes $= 24^2$

$\therefore probability = \frac{320}{576} =\frac{5}{9} > 50\% $

[Anindya Biswas]

I have another solution,

Let's make a $24\times24$ grid and we will write $max(i, j)$ in the cell $(i, j)$ which is the intersection of $i$th column and $j$th row. A similar $5\times5$ grid would look like this:

A $5\times5$ grid for similar problem

IMG_20210209_003644.jpg (22.67KiB)Viewed 3928 times

Now it should be clear that $Pr\left(max(i, j)\geq N\right)=1-\frac{(N-1)^2}{24^2}$
Now, $1-\frac{(N-1)^2}{24^2}>\frac12\Rightarrow N<12\sqrt2+1$
And greatest such integer is $17$.

[Mehrab4226]

I have another solution,

Let's make a $24\times24$ grid and we will write $max(i, j)$ in the cell $(i, j)$ which is the intersection of $i$th column and $j$th row. A similar $5\times5$ grid would look like this:IMG_20210209_003644.jpg

Now it should be clear that $Pr\left(max(i, j)\geq N\right)=1-\frac{(N-1)^2}{24^2}$
Now, $1-\frac{(N-1)^2}{24^2}>\frac12\Rightarrow N<12\sqrt2+1$
And greatest such integer is $17$.

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

[Anindya Biswas]

I have another solution,

Let's make a $24\times24$ grid and we will write $max(i, j)$ in the cell $(i, j)$ which is the intersection of $i$th column and $j$th row. A similar $5\times5$ grid would look like this:IMG_20210209_003644.jpg

Now it should be clear that $Pr\left(max(i, j)\geq N\right)=1-\frac{(N-1)^2}{24^2}$
Now, $1-\frac{(N-1)^2}{24^2}>\frac12\Rightarrow N<12\sqrt2+1$
And greatest such integer is $17$.

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

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.

[Shon4poth-সঞ্চারপথ]

I have another solution,

Let's make a $24\times24$ grid and we will write $max(i, j)$ in the cell $(i, j)$ which is the intersection of $i$th column and $j$th row. A similar $5\times5$ grid would look like this:IMG_20210209_003644.jpg

Now it should be clear that $Pr\left(max(i, j)\geq N\right)=1-\frac{(N-1)^2}{24^2}$
Now, $1-\frac{(N-1)^2}{24^2}>\frac12\Rightarrow N<12\sqrt2+1$
And greatest such integer is $17$.

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]

I have another solution,

Let's make a $24\times24$ grid and we will write $max(i, j)$ in the cell $(i, j)$ which is the intersection of $i$th column and $j$th row. A similar $5\times5$ grid would look like this:IMG_20210209_003644.jpg

Now it should be clear that $Pr\left(max(i, j)\geq N\right)=1-\frac{(N-1)^2}{24^2}$
Now, $1-\frac{(N-1)^2}{24^2}>\frac12\Rightarrow N<12\sqrt2+1$
And greatest such integer is $17$.

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 $N<24$

We are provided that $N\leq24$. $Pr\left(max(i,j)\geq N\right)=1-Pr\left(max(i,j)\in\{1,2,3,\dots,N-1\}\right)=1-Pr\left(i,j\in\{1,2,3,\dots,N-1\}\right)$
That's why it is $(N-1)^2$