ball and the numbers?

Question

There are four balls, each with a distinct integer label, in a bag. It is known that the product of the
integers on the labels is 24. If a ball is picked at random, the probability that the label is an even natural
number is:
(1) ¼ (2) ¾ (3) ½ (4) None of these

solution:
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If there are four balls in a bag each with an integer label and the product = 24, then the labels have to be 1, 2, 3,4 or (-1, -2, -3, -4) or (1, 2, -3, -4) or (1, -2, -3, 4) and so on. As the product of the 4 integers is positive either two
or four of the integers have to be positive. Taking conditional probabilities, the answer obtained is none of the
choices (1) to (3).


Question:
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i dont understand how they solved it. whats the meaning with "Taking conditional probabilities, ......"

how this can help to achieve the solution ?

can u help ?

thanks

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i thought the solution would be 1/2.

my logic ={posible case}/all favourable case = 2/4=1/2


but my book has given the solution like that which is confusing.

Answer 1

(1, 2, 3,4) or (-1, -2, -3, -4) or (1, 2, -3, -4) or (1, -2, -3, 4)
You also have
(-1, -2, 3, 4), (-1, 2, -3, 4), (-1, 2, 3, -4), and (1, -2, 3, -4)
So there are 8 possibilities.
Each having a probability of 1/8 or occurring.
For (1,2,3,4) and (-1,2,-3,4), the prob. of an even natural number is 2/4 = 1/2
Multiply this by 1/8 to get the conditional prob.
(1/2)(1/8) = 1/16
For each of the following: (1,2,-3,-4), (1,-2,-3,4), (-1, -2, 3, 4), and (-1, 2, 3, -4), the prob. of an even natural number is 1/4
(1/4)(1/8) = 1/32
For (-1,-2,-3,-4) and (1,-2,3,-4), the prob. of an even natural number is 0.
0(1/8) = 0
Add all these together.
2(1/16) + 4(1/32) + 2(0)
= 1/8 + 1/8
= 2/8
= 1/4
This doesn't match your solution, but you can see if this makes any sense to you.

Answer 2

The four numbers have to multiply to 24
1*2= 2....2*3=6....6*4=24
However, if all these numbers were negative you would still get possitive 24 because 2 negatives cancel out to make a positive.
Also, if only 2 numbers were negatives they would cancel out and the answer would still be positive.
Therefore, so long as the number of negative numebrs is even, the numbers 1, 2, 3, 4 work
A natural number is a positive integer
If you have four negatives, the probability is zero
If you have 2 even positives or all positives, the proabaility is 2
If you have only two odd positives the probability is zero

Answer 3

This seems more difficult because the balls were labelled with integers. If it would have been natural numbers then the answer would have been easy to find. I don't know much about probablity.

All I can say that according to me the answer should be 1/2