A bag contains a total of four billiard balls. The billiard

In the first case we consider (2,3),(2,5),(2,7) but why do we not consider (3,2), (5,2), (7,2). But then total no. of cases will be 6 and Probability cannot be 6/6 = 1. I m too confused Help!!!

Yes, that is confusing. But consider this:

Scenario 1: P(sum odd)=Favorable outcomes/Total # of outcomes
Total # of outcomes=4C2=6, # of selection of 2 balls out of 4 balls.
Favorable outcome: two numbers can total odd if one number is even 2 and another odd 3,5,7. So two ball must be (2,3),(2,5),(2,7). So we have 3 favorable outcomes.

P(sum odd)=Favorable outcomes/Total # of outcomes=3/6=1/2

Favorable outcome is the PAIR of balls, therefore it doesn't matter it'll be (2,3) or (3,2), it's still ONE pair. On the other hand for the total # of outcomes we are also counting number of drawing of TWO balls out of 4.

BUT we can count this in another way (the way we did for scenario 2):
First ball 2, seond any odd =1/4*3/3(as after drawing the even ball there are 3 balls left and all are odd)=1/4
First ball any odd, second ball 2 =3/4*1/3(as after drawing the odd ball there are 3 balls left and only one is even)=1/4.
Final P=1/4+1/4=1/2

OR: 2*1/4*3/3=1/2



Scenario 2: Different situation. But again the odd sum can occur if either of drawing gives even 2 and another odd 3,5,7.

This can happen in TWO ways:
even (2), odd (3,5,7); OR
odd(3,5,7), even(2).

P(sum odd)=2(as we have 2 ways favorable sum can occur)*1/4(probability of picking even 2, 1 ball out of 4 balls)*3/4(probability of picking odd 3,5 or 7, 3 out of 4 balls)=2*1/4*3/4=3/8


We also can count the probability here with combination:
2*1C1*3C1/4C1*4C1=2*(1*3)/(4*4)=3/8

1C1=# of selections of even out of one
3C1=# of selections of odd out of three
4C1=TOTAL # of selections of first ball out of 4
4C1=TOTAL # of selections of second ball out of 4 (as after drawing the first one we put it back, so there are still 4 balls in the bag)
Multiplying this by 2 as there are two ways: ood/even or even/odd.

Hope it's clear.

I like this method more. And i know u get this a lot these days but this is awesome stuff.. Thnks a ton..

1 3/6 = 1/2

2 let 1st ball be 2 then 2nd ball has to be 3,4,5 thus probability = 1/4 * 3/4

let 1st ball be either 3,4 or 5 and 2nd ball be 2 then = 3/4 * 1/4

thus 2* 1/4 * 3/4

Scenario 1:

1 ball has to be 2

(1 * 3)/(4C2) = 3/6 = 1/2

Scenario 2:

P(2)*P(any other number) + P(Any other number) * P(2)

= 2 * 1/4 * 3/4

= 6/16

= 3/8

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