We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
247
2
avatar+630 

A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction.

 Dec 22, 2018
 #2
avatar+5074 
+2

\(\text{break the figure into 3 pieces}\\ \text{the first piece is 2x1001 white squares to the left of the shaded squares}\\ \text{the second piece is the 2 shaded squares}\\ \text{the third is the 2x1001 white squares to the right of the shaded squares}\\ \text{to pick a rectangle that does not contain the shaded squares we pick two points}\\ \text{from either the first section or the third section and these define our rectangle}\\ \text{there are }\dbinom{1001}{2} =500500\text{ ways to pick 2 points in both sections}\\ \text{so there are }2 \cdot 500500 = 1001000 \text{ ways to pick the rectangles that contain no shaded squares}\)

 

\(\text{There are }\dbinom{2003}{2} = 2005003 \text{ total ways to pick 2 points from the 2003}\\ \text{thus }\\ P[\text{rectangle contains no shaded square}]= \dfrac{1001000}{2005003 } = \dfrac{1000}{2003}\approx 0.4992\)

.
 Dec 22, 2018

10 Online Users

avatar