Probability Question

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mattgad

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Oct 21, 2005
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Of the households in Edinburgh, 35% percent have a freezer, and 60% have a colour TV. Given that 25% of households have both, calculate the probability that a household has either a freezer or a colour TV but not both.

The answer is 0.45. I am using the following formula.

P(TV u F) = P(TV) + P(F) - P(TV n F)

I am using P(F) = 0.1, as 0.35 is for a freezer, but 0.25 is for people with both. I am also using P(TV) = 0.35, as a TV is 0.60 percent change, so 0.25 is for both.

I am also using P(TV n F) = 0.25

Where am I going wrong?

Thanks
 
Use the notation \(\displaystyle A^c\) to mean the complement of A.

Here is what you are asked, have exactly one: \(\displaystyle P(TV^c \cap F) + P(TV \cap F^c )\)

The following is true: \(\displaystyle P(TV) = P(TV \cap F) + P(TV \cap F^c )\).
Using that we get \(\displaystyle P(TV \cap F^c ) = .25\).

Now you finish.
 
Yes, but you don’t need to use that.
Just find \(\displaystyle P(TV^c \cap F)\) and add two together. Your done!
 
I thnk you could solve it much easier if you create Venn diagrams. Create two circles intersecting each other like olympic rings. One circle is fridge owners and other TV owners. The common part between the two circles would be fridge and TV owners.
This way you can solve many different probability values wihout actually going into the rut of formulas n stuff
 
questionvenn1kt.png


I have drawn it out as a Venn Diagram.

Have I done it correctly?

If so,

\(\displaystyle P(F u TV) = P(F) + P(TV) - P(F n TV)\)
\(\displaystyle P(F u TV) = 0.1 + 0.35 - 0.25 = 0.2\), which is incorrect.

Edit: Just realised in the image, TV and Fridge are labelled the wrong way round.
 
You did the Venn diagram correctly:
Read what the question asked: a household has either a freezer or a colour TV but not both. Owns one but not the other. That is exactly one of the two. That is the symmetric difference of the sets.
Now add the indicated numbers.
venn22do.gif


P.S. Forget the labels. They do not matter.
 
The diagram is correct except the .05. When you add them all up the probaility MUST = 1.00
.1+.25+.35 = .7
That leaves .3 for the outside of the circles (those who have neither..
.1+.35=.45 who have one but not both.
 
What is the probability that of all households in Edinburgh at least the TV or Freezer has been stolen in the past year :D :D
 
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