Log, floor and ceiling together will make you surrender!

Probability Level 5

Given that $x$ is a random real number between 1 and 4 and $y$ is a random real number between 1 and 9.

The expected value of: $\lceil \log_{2}x \rceil- \lfloor\log_{3}y \rfloor$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are positive and relatively coprime integers. Then, find the value of $100m+n$ .

The answer is 1112.

1 solution

Pranjal Jain
Dec 10, 2014

Graph Graph

Let $f(x,y) = (log_{2} x)-[log_{3} y]$

$A_{1}:$ Area= $2$ and $f(x,y)=1$

$A_{2}:$ Area= $4$ and $f(x,y)=2$

$A_{3}:$ Area= $6$ and $f(x,y)=0$

$A_{4}:$ Area= $12$ and $f(x,y)=1$

Hence, expected value of $f(x,y)=\dfrac{(2×1)+(4×2)+(6×0)+(12×1)}{2+4+6+12}=\dfrac{11}{12}$

What is the term expected value?

Mayank Chaturvedi - 6 years, 1 month ago

The most probable value according to probability. For example, say, you'll get one cake if a head appears and two cakes if a tail appears in a coin toss. Then expected number of cakes you'll get is $1.5,\quad 1\times 0.5+2\times 0.5$

Pranjal Jain - 6 years, 1 month ago

That is a new concept for me. Understanding a bit : )

Mayank Chaturvedi - 6 years, 1 month ago

Log, floor and ceiling together will make you surrender!

The answer is 1112.

1 solution

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