Probability...........?

Question

In a street containing 20 houses, 3 households do not own a television set; 12 household have a black and white set and 7 household have a colour and a black and white set. Find the probability that a household chosen at random owns a colour television set,

The answer is 3/5

The question is so confusing, I don't understand a thing x.x

Answer 1

It's not your fault that you are confused.  If the information that you've supplied is ALL the information given in the question, then it has allowed for some interpretation.  For example, you've stated that 3 households do not own a television, but if IN FACT 4 households do not have one, then it's true that 3 do not, as well...

Let's break this down and look at it graphically.  I'll list out 20 houses and the information that we have about each one:

1)    no tv    

2)    no tv    

3)    no tv    

4)    bw        

5)    bw        

6)    bw        

7)    bw    

8)    bw        

9)    bw/c    

10)  bw/c    

11)  bw/c

12)  bw/c

13)  bw/c

14)  bw/c

15)  bw/c

16)  no info

17)  no info

18)  no info

19)  no info

20)  no info

If we interpet the statements as:

A) exactly 3 of the twenty (no more and no less) have zero tv sets,

B) 12 houses (no more and no less) have a black and white,

C) 7 (of the 12 with black and white sets) also have a color set,



then it follows from statement A that the remaining houses MUST have a tv, and it follows from B that it's not a black and white.  Therefore the remaining households have color tv sets.

Counting them up, there are 12 households with color tv's (7 of which have both a bw and color, and 5 of which we were given no information, but we concluded had a color tv), and 12/20 is 3/5.

QED.

However....

If the statement (A) is interpreted to mean "in particular there are 3 houses without a tv, but it could be 4 houses or 5 houses or whatever more than 3...", then the remaining 5 houses may have anywhere between zero and 5 color tv's, and the total number of color sets in the twenty houses could be anywhere between 7 and 12 total.

The thing about logic puzzles is that the details mean EVERYTHING.  The way that the writers of logic puzzles can trick you is by making you incorrectly assume something because of the wording of the question.  In this case, it's a simple math question, but the answer hinges on you making a logical conclusion about the remaining households on which no information is given.  Because the statements that you provided do no solidly lock down the status of those remaining houses, there is ambiguity.

A puzzle in which the information provided is insufficient is poorly designed.  We describe it as being an "ill posed" question.

This question was ill posed.

Your confusion is not your fault.

Answer 2

U said that 3 households do not have a TV set

It means 17 do have  colour or black and white

12 households have a black and white TV set

It means 5 have only a colour TV

7 households  have a colour and black and white

Ir means that 5+7=12 households own a colour TV

(5 only a colour TV and 7 who have both ones)

So 12 out of 20 have a colour TV set!

Probability that a household randomly chosen

own a colour TV set is 12/20 = 3/5

Answer 3

Isn't the answer supposed to be 7/20?

Answer 4

Did you learn about Venn diagrams?  It would be helpful.

Since 7 homes have both color and BW, then 5 homes must have BW only.  (A total of 12 households have BW.  7 have both, so 12 - 7 = 5 have BW only)

Now, 3 households do not own a TV, so

3 (no TV) + 5 (BW only) + 7 (both color and BW) = 15

which leaves 5 homes with only a color set, since there are 20 houses on the block.

Probability of owning a color set = Homes with color TV/all homes

P(color) = 12/20 = 3/5