Probability question

Question

Survey taken of 1600 cat owners found following:

685 cats were male

398 had long hair

417 were black

205 were male and had long hair

216 were male & black

94 had long hair & were black

59 were all three (male, long haired & black)

 

a) How many cats were male and black, but did not have long hair?  (I'm getting 216 but not sure that's right)

 

b) How many cats were not male, did not have long hair. and were not black?  (If 1600 were surveyed and 685 were male, that leaves 915 female, so I know I would start with 915, but get a bit lost after that).

Answer 1

In questions like this, it's important to make a distinction between "male and black" and "male and black without long hair." The first category, "male and black," specifically includes male black cats with long hair and male black cats without long hair. There are 216 of these cats, as you're told in the problem. The second category, "male and black without long hair," includes only male black cats without long hair. Keep an eye out for unstated distinctions like these.

 

The best way to solve this type of problem is to draw a Venn Diagram. Draw three circles overlapping and label one M (for male), one B (for black), and one L (for long-haired).

 

It's a good idea to start in the middle of the Venn Diagram. In this case, that's the section where all three circles overlap. This represents all cats who were male, long-haired, and black. We're told there are 59 of them, so label that section 59.

 

Now we can work our way out. Let's fill in the section where M and L overlap. There are a total of 205 cats that were male with long hair, but that includes the 59 that were also black. So, subtract 59 to get 205 - 59 = 146 cats in this section.

 

We'll do the same for M and B (216 - 59 = 157) and L and B (94 - 59 = 35).

 

Now we'll work one step further out. There are 685 male cats, but that includes male and black cats, male and long-haired cats, and cats who are all three. So to figure out what to put in the last remaining sector of the M circle (where cats must be male but not black or long-haired), we'll need to subtract the male and black, male and long-haired, and all three cats: 685 - 146 - 157 - 59 = 323 cats in this section.

 

Similarly, there are 398 - 146 - 35 - 59 = 158 long-haired cats and 417 - 157 - 59 - 35 = 166 black cats.

 

Finally, let's figure out how many cats aren't included in our Venn Diagram so far. These are the cats that are not black, not male, and not long-haired. We'll add up all the numbers we've already written down to figure out how many cats have at least one of the characteristics, then subtract that from 1600 to find out how many cats had none of the characteristics: 1600 - (323 + 146 + 158 + 157 + 59 + 35 + 166) = 556 cats that were female, short-haired, and not black.

 

 

 

Hopefully that process makes sense! Remember to start from the middle of the Venn Diagram and don't mix up "A and B" with "A and B but not C."

 

Now, you can just read your answers from the Venn Diagram:

 

a) 157 cats are male and black without long hair

 

b) 556 cats are not male, not long-haired, and not black