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very simple probablility question...

S

Semidevil

Diamond Member
ok, so here is this simple problem.

given a box w/ coins labeled 1 - 24, what is
P(a): probability of getting coins divisible by 2,
P(b): picking coins divisible by 3.
P(a intersect b)
P(a union b).

ok, I know to find P(a) and P(b). very simple. As for P(a intersect b) and P(a union b), I"m getting confused. I can do it conceptually, by just counting counting...but how do I do it mathematically?

I always thought that P(a union b) = p(a) + p(b), but then, I see p(a union b) = p(a) + p(b) - p(a intersect b) at another section.

And then for P(a intersect b), I always thought it was = to P(a) * p(b), but my book says, p(a intersect b) = P(a) +p(b) - p(a union b). This then requires me to find the union first.

drawing the venn diagrams, the formulas makes sense, but when I count coin by coin, it doesnt match up.
 
P(a) is 1/2

P(b) is 1/3

P(a intersect b) = 1/2 * 1/3 = 1/6

P(a union b ) = 1- (1-1/2) *(1-1/3) =2/3



The union equation of p(a) + p(b) - P(a intersect b) is also correct.
 
P(a intersect b) = P(a) * P(b) is definitely correct

however,

P(a union b) != P(a) + P (b) .

you can get the union 2 different ways. One is with the P(a) + P(b) - P(a intersect b) equation.

The other way I listed is that you find P( !a intersect !b) and then subtract that from 1.
 
p(a) = 100%
p(b) = 100%

all numbers are divisible by 2 and 3. *Evenly* divisible by is another matter.

Take a look at a Venn Diagram to see why those equations are correct.
 
Originally posted by: Semidevil
ok, so here is this simple problem.

given a box w/ coins labeled 1 - 24, what is
P(a): probability of getting coins divisible by 2,
P(b): picking coins divisible by 3.
P(a intersect b)
P(a union b).

ok, I know to find P(a) and P(b). very simple. As for P(a intersect b) and P(a union b), I"m getting confused. I can do it conceptually, by just counting counting...but how do I do it mathematically?

I always thought that P(a union b) = p(a) + p(b), but then, I see p(a union b) = p(a) + p(b) - p(a intersect b) at another section.
Click to expand...
The reason for the extra subtraction is that P(a) + P(b) counts the intersection twice. You must subtract out the extra portion that is counted twice
And then for P(a intersect b), I always thought it was = to P(a) * p(b), but my book says, p(a intersect b) = P(a) +p(b) - p(a union b). This then requires me to find the union first.

drawing the venn diagrams, the formulas makes sense, but when I count coin by coin, it doesnt match up.
Click to expand...
Both are right.

p(a) = {2,4,6,8,10,12,14,16,18,20,22,24} = 12/24
p(b) = {3,6,9,12,15,18,21,24} = 8/24
p(a intersect b) = {6,12,18,24} = 4/24
p(a union b) = {2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24} = 16/24

Notice that in p(a) + p(b) you are counting 6, 12, 18,and 24 twice? That is why you subtract out the intersection.


 
You're right, in this example to find P(a intersect b), you'll end up counting and there's not much of an equation to get it. Then from that you can find P(a union b). Oh, but for future information... here's another formula.

P(a intersect b) = P(a) * P(b given a)

 
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