Question 1162354
{{{drawing(300,200,-4,4,-2,4.8,
rectangle(-4,-1.6,4,4.4), locate(-2,1.8,a),locate(1.5,1.7,c),
locate(-3.7,-1,d),
 locate(-3.7,4,Botany), locate(-.1,1.8,b),
red(circle(-sqrt(2),sqrt(2),2)),
red(circle(-sqrt(2),sqrt(2),1.95)),
red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),
locate(2.5,4,Biology)
 )}}} 



Among the class of 100 students,<pre> 
So a+b+c+d = 100
</pre>75 studied Botany,<pre> 
So a+b = 75
</pre>55 studied Biology<pre> 
So b+c = 55
</pre>and 40 students studied both of the subjects.<pre> 
So b=40

Substituting 40 for b in b+c = 55
                        40+c = 55
                           c = 15

Substituting 40 for b in a+b = 75
                        a+40 = 75
                           a = 35

Substituting a=35, b=40, and c=15 in
                     a+b+c+d = 100
                  35+40+15+d = 100
                        90+d = 100
                           d = 10

{{{drawing(300,200,-4,4,-2,4.8,
rectangle(-4,-1.6,4,4.4), locate(-2,1.8,35),locate(1.5,1.7,15),
locate(-3.7,-1,10),
 locate(-3.7,4,Botany), locate(-.1,1.8,40),
red(circle(-sqrt(2),sqrt(2),2)),
red(circle(-sqrt(2),sqrt(2),1.95)),
red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),
locate(2.5,4,Biology)
 )}}}
</pre>A student is selected at random from this class what is the
probability that
i) student studied at least one subject<pre>
All but the 10.  So that's 100-10 = 90 out of 100, so the probability
is 90/100 which reduces to 9/10
</pre>ii) student studied Botany but not Biology<pre>
That's the 35.  So that's 35 out of 100, so the probability
is 35/100 which reduces to 7/20
</pre>iii) student studied Biology but not Botany.<pre>
That's the 15.  So that's 15 out of 100, so the probability
is 15/100 which reduces to 3/20
</pre>iv) student studied none of the subjects<pre>
That's the 10 that aren't in either circle.  So that's 10 out of 100, 
so the probability is 10/100 which reduces to 1/10.

Edwin</pre>