Question 1134540
400 students are surveyed about the pets they have. 250 have a dog, 190 have a
cat and 160 have a dog and a cat. What is the probability that a student has a
dog?

<pre>
The probability of a dog is 250 out of 400 = 250/400 = 5/8

For the other question I think a Venn diagram is better.

Let D = the set of students who own dogs.
Let C = the set of students who own cats.
Let w = the number of students with dogs but NO cats.
Let x = the number of students both cats and dogs.
Let y = the number of students with cats but NO dogs. 
Let z = the number of students with NO dogs and NO cats.

{{{drawing(300,200,-4,4,-2,4.8,

rectangle(-4,-1.6,4,4.4), locate(-2,1.8,w),locate(1.5,1.7,y),

locate(-3.7,-1,z),

 locate(-3.6,2.5,D), locate(-.1,1.8,x),

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(3.4,2.5,C)
 )}}}

</pre>400 students are surveyed about the pets they have.<pre>

So w + x + y + z = 400.  

</pre>250 have a dog,<pre>

So w + x = 250

</pre>190 have a cat,<pre>

So x + y = 190

</pre>160 have a dog and a cat.<pre>

So x = 160

Substitute 160 for x in w + x = 250
                      w + 160 = 250
                            w = 90

Substitute 160 for x in x + y = 190
                      160 + y = 190
                            y = 30

Substitute w=90, x=160, y=30, in w + x + y + z = 400
                             90 + 160 + 30 + z = 400
                                       280 + z = 400
                                             z = 120

[You weren't asked for z, but you should know how to find it anyway,
in case you'd have been asked about petless students]

{{{drawing(390,260,-4,4,-2,4.8,

rectangle(-4,-1.6,4,4.4), locate(-2,1.8,w=90),locate(1.5,1.7,y=30),

locate(-3.7,-1,z=120),

 locate(-3.6,2.5,D), locate(-.45,1.8,x=160),

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(3.4,2.5,C)
 )}}}

</pre>And what is the probability the student has a dog OR a cat?<pre>

That's the number in both circles over 400, or (w+x+y)/400 = (90+160+30)/400 =
280/400 = 7/10

Edwin</pre>