OG 11 (Green Book) #151

[mdavis]

Hi, I am having a problem understanding part of the explanation in the OG. Could someone clarify this for me? Here is the question:

A box contains 100 balls, numbered from 1 to 100. If three balls are selected at random and with replacement from the box, what is the probability that the sum of the three numbers on the balls selected from the box will be odd?

a) 1/4
b) 3/8
c) 1/2
d) 5/8
e) 3/4

So the guide says to find out the different probabilities for the different ways it could add up. So you end up with 1/8, 1/8, 1/8, and 1/8 for each of the ways it could be added. I get that.

What I don't understand is when the guide then says to multiply 1/8 by 4. I mean I understand visually since I have 4 things here, but I don't understand in terms of why I would multiply by 4 instead of adding 1/8 4 times or multiplying 1/8 by 1/8 by 1/8 by 1/8?

I understand that if you have a coin that is heads or tails and there are two tosses, then the probability is 1/2 multiplied by 1/2 for the two outcomes for two heads in a row or tails in a row, which is 1/4. It seems to me like the OG is saying instead of doing, in my example, 1/2 times 1/2, just do 2 times 1/2.

Anyways, I don't understand that last part and any help would be appreciated. Thanks!

[vivek.kapoor83]

sol is
There are 50 Odd no of balls and 50 even num balls
So, there will be cases

odd,odd,odd = odd => 1/2*1/2*1/2
odd,even,even = odd => 1/2*1/2*1/2
even ,odd,even = odd => 1/2*1/2*1/2
even,even,odd = odd => 1/2*1/2*1/2

So, Overall prob = 4(1/8) = 1/2


Where you see "OR" ..u add prob..coz overall prob > Indv prob
Where you see "AND" ...u multiply prob. coz Overall prob < Indv prob.

[willbeatthegmat]

there are 4 ways in which u can pick up the number..i.e..either u pick:
1)odd+ odd+ odd = odd
2)odd +odd + even = even
3)even+ even+ odd = odd
4)even+even+ even = even

the probability of pickin any of these cases is equal....and in total there r 4 cases out of which 2 results in odd...so 2/4 i.e. 1/2 wud b the ans