The way I interpreted the problem:
We are given that each of the 3 integers in each set are distinct and POSITIVE Integers.
The questions asks us: “what is the probability that: RS = R?”
Or
RS - R = 0
R (S - 1) = 0
Zero Product Rule: either R = 0 or S = 1
Since we have only positive integers, the only way for this to be possible is if S = 1.
Since all 3 integers WITHIN a Set must be distinct, there are 2 possibilities:
Either: (case 1) +1 is one of the three Distinct positive integers in Set S
Or: (case 2) +1 does NOT exist in Set S
It does not matter which of the 3 positive integers we pick from Set R. All that matters, in order to get the favorable outcome, is that we pick the +1 from Set S.
Case 1: Set S {1 , A , B} —— where A and B are distinct and do not equal 1
Probability that (RS = R) = (1/3)
Or
Case 2: Set S {X , Y , Z} ——- where X, Y. Z are distinct positive integers and NONE of them is equal to +1
Probability that (RS = R) = 0 (no chance)
In other words, the contents of Set R are irrelevant. We want to know whether Set S contains +1 as one of its three distinct positive integers.
S1: tells us that Set R must contain +1
For RS = S, given that these are all positive integers —— R must = +1
So set R contains {1 , P, Q} —- where P and Q are distinct positive integers that do not equal +1
We can select any number from Set S (3/3)
And
Probability of selecting the 1 from set R is = (1/3)
Hence, probability that RS = S is (1/3)
Does not tells us whether Set S does or does NOT contain +1
Not sufficient.
Statement 2:
For R + S = 2 ——- given that R and S each must be Positive Integers, there is only one possibility:
R = 1 —— and —— S = 1
Since the probability that R + S = 2 is not zero, we know that Set S MUST contain +1 as one its three distinct positive integers.
And the probability of pulling a +1 from set S is therefore = (1/3)
Which means the probability that RS = R is also (1/3)
Statement 2 is sufficient alone.
B
Rs1991 wrote:
Sets R and S each contain three distinct positive integers. If integer r is randomly selected from R and integer s is randomly selected from S, what is the probability that rs = r?
(1) The probability that rs = s is 1/3
(2) The probability that r + s = 2 is 1/9
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