Probability

The numbers ?1, ?2, ?3 and ?4 are drawn one at a time from the set {0, 1, 2,…, 9}. If these four numbers are drawn with replacement, what is theprobability that ?1?4 − ?2?3 is an even number?

Options :

A) 1/2
B) 1/4
C) 3/8
D) 3/4
E) 5/8

Since it is with replacement, I thought it would be 1/2 but I am not sure.

Firas said:

The numbers ?1, ?2, ?3 and ?4 are drawn one at a time from the set {0, 1, 2,…, 9}. If these four numbers are drawn with replacement, what is the probability that ?1?4 − ?2?3 is an even number?

Options :

A) 1/2
B) 1/4
C) 3/8
D) 3/4
E) 5/8

Since it is with replacement, I thought it would be 1/2 but I am not sure.

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I would start by thinking about the conditions under which ?1?4 − ?2?3 is even. That will happen if either ?1?4 and ?2?3 are both even, or if they are both odd. In turn, ?1?4 and ?2?3 are both even when either ?1 or ?4 is even, and ?2 or ?3 is even. Find the probability of that. And so on.

The answer is not 1/2.

I got the answer.

odd x odd - odd x odd = even

odd x even - odd x even = even

even x even - odd x even = odd

odd x odd - even x even = even

odd x even - even x even = even

even x even - even x even = even

even x odd - odd x odd = odd

odd x odd - odd x even = odd

Therefore , 5/8 are even.

Firas said:

odd x odd - odd x odd = even

odd x even - odd x even = even

even x even - odd x even = odd

odd x odd - even x even = even

odd x even - even x even = even

even x even - even x even = even

even x odd - odd x odd = odd

odd x odd - odd x even = odd

Therefore , 5/8 are even.

Click to expand...

Your answer is correct, but you've only listed 8 of the 16 possibilities, so you haven't justified your answer. Why did you do this? What order did you use in listing them, to make sure you didn't miss any? I see no rhyme or reason to what you did. I would have expected something orderly like this:


I did it without listing the sample space. The probability that both products are even is Pr(not both of first pair are odd)*Pr(not both of second pair are odd) = (1 - 1/4)*(1 - 1/4) = 3/4 * 3/4 = 9/16; the probability that both products are odd is Pr(both of first pair are odd)*Pr(both of second pair are odd) = 1/4 * 1/4 = 1/16. The sum is 10/16 = 5/8.

thanks!