Hitting a square dartboard

Probability Level 2

ABCD is a square shaped board. 4 equal rectangles are drawn into it. The length of the sides of the rectangles are x x and y y where x / y = 3 x/y = 3 . Rahat throw a dart to the board. If the probability of the dart hitting the black portion is a / b a/b where a , b a, b are coprime positive integers, then find a + b a + b .


The answer is 5.

Akash Hossain by Akash Hossain , 18, Bangladesh

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1 solution

Given x y \frac{x}{y} = = 1 3 \frac{1}{3} , we realize that:

The side of the bigger square = = 4 y 4y and the black figure thus formed is a square with each side = = 2 y 2y

So, P ( Hitting the black figure ) = A r . I n n e r S q u a r e A r . A B C D = ( 2 y ) 2 ( 4 y ) 2 = 1 4 P(\text{Hitting the black figure}) = \frac{Ar. Inner Square}{Ar. ABCD} = \frac{(2y)^{2}}{(4y)^{2}} = \frac{1}{4}

Hence, the solution sought = 1 + 4 = 5 = 1 + 4 = \boxed{5}

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But what will be the explanation of using area in counting the probability?

Akash Hossain - 3 years, 4 months ago

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Area of ABCD denotes all possible outcomes - all the points the dart can hit will be confined within ABCD.

Area of inner square denotes all favourable outcomes - all the points the dart needs to hit will be confined within this inner square.

A Former Brilliant Member - 3 years, 4 months ago

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