Probability Question - 1 - x trick vs. straight mult?

[Eagles121882]

Hi, I'm really bad with probability questions and need some clarification. Here's a made up scenario demonstrating my confusion: Let's say you have 6 different sized rocks and you pick two rocks. What is the probability that you pick the smallest rock first and then the largest rock second in two tries?

If I use the 1 - x (probability of not picking) the equation would be: 1 - prob of not picking smallest rock (5/6) x prob of not picking largest rock (4/5) = 1 - 20/30 = 1/3. Is that correct?

vs.

If I do straight multiplication: prob of picking smallest rock (1/6) x prob of picking largest rock (1/5) = 1/30.

I know I'm missing concepts somewhere but I'm not sure where. Can someone please help show me which is the correct answer in this scenario and I think that would help my understanding.

Thanks!

(Sorry, I realized after posting that this question should prob be in the general math section but doesn't seem like I can move the post to that section now.)

[atul.prasad]

Hi , I guess your second approach is correct.

Intuitively too, 1/3 should be a wrong answer.
Why?

The question asks you to select rocks in a particular order.
Lets number the rocks 1 to 6 in the order smallest to largest. Then the order (1,6) is the only one pair that qualifies out of all possible pairs. An answer of 1/3 means 1 in 3 picks would be (1,6). Does that seem reasonable ? I guess not :)

There are 6P2 ways to choose the rocks (since order matters) = 30.
And we know that it is only the order 1,6 that satisfies the condition "smallest rock first, largest rock second". So the probability is 1/30

[RonPurewal]

If I use the 1 - x (probability of not picking) the equation would be: 1 - prob of not picking smallest rock (5/6) x prob of not picking largest rock (4/5) = 1 - 20/30 = 1/3. Is that correct?

nope -- in fact, there are multiple mistakes in this approach.

the first mistake is that you are not including the correct scenarios in your "opposite" case.
i.e., you have characterized the opposite case as picking NEITHER of these two rocks. however, that's not necessary: you can have ANY situation that is not "smallest rock first, largest rock second". in other words, it's perfectly okay to pick the smallest rock first, as long as you don't pick the largest rock second; and vice versa.
so you have to split this up into cases:
a * first rock NOT smallest, second rock largest;
b * first rock NOT smallest, second rock NOT largest;
c * first rock smallest, second rock NOT largest.
as for the first two cases -- together, these cases cover all of the possible outcomes in which the first rock is NOT the smallest. so the probability of these cases together (a and b) is just the probability that the first rock is not the smallest, i.e., 5/6.
then you have to add on the probability of case c, which is (1/6)(4/5) = 4/30.
so all of your UN-wanted cases total 5/6 + 4/30 = 25/30 + 4/30 = 29/30.
therefore, the probability of the desired event is "1 - x" = 1 - 29/30 = 1/30.

--

the second mistake, by the way, is in your calculation of the event "first rock not smallest, second rock not largest" -- this event is actually more complicated than you made it out to be; it's not just 5/6 x 4/5.
the reason is that the second probability (the one you've written as 4/5) isn't always 4/5; it actually depends on the outcome of the first choice. see, the first rock has to be "not smallest" -- but what you didn't realize is that this first rock could in fact be the largest one.
so you would have to separate this event into two possibilities, one in which the first rock chosen is actually the largest one (in which the second probability would be 5/5 rather than 4/5, since it would be impossible to choose the largest rock in that scenario), and the other in which the probability is 4/5 as you've written.

vs.

If I do straight multiplication: prob of picking smallest rock (1/6) x prob of picking largest rock (1/5) = 1/30.

this is obviously the better way to do this problem.

[Eagles121882]

Got it. Thanks for the clarification!

[jnelson0612]

Excellent Ron--thank you!