A jar contains 8 red marbles and y white marbles.

[karizmaticafroz]

A jar contains 8 red marbles and y white marbles. If Joan takes 2 random marbles from the jar, is it more likely that she will have 2 red marbles than that she will have one marble of each color?

(1) y ≤ 8
(2) y ≥ 4

Source: DS question mgmat - test2
The OA is B( Statement 2 alone)
The question asks to find if the probability or RR is greater than RW(WR)... But the explanation takes the selection process RW and WR in the second event as 2 different cases and multiplies 2 with the reduced form of the question.
The probability of obtaining one red marble and one white marble is given by:
P(Red AND White) = (8/8+y) * (y/7+y) * 2 = (16y/(8+y)(7+y))

Which I doubt is the right way. The question clearly asks to find the probability of which COMBINATION is greater. Why should we consider the permutation (RW, WR) ?
So according to my explanation, the question will be reduced to 7>y ?
Then the answer should be E. Please explain why we consider the order (RW,WR) ?

[RonPurewal]

Two ways to think about this.

1/
Imagine that you're picking the two marbles out one at a time.
In this case, it should be obvious that getting a red marble first, and then a white marble, is different from getting a white marble first and then a red one.
By contrast, there's still only one way to get a white marble and then another white marble. There's also only one way to get a red one and then another red one.

The probabilities aren't going to change if we decide to pick the marbles out one at a time, so the same applies to the simultaneous choice.

2/
Just check the numbers. Calculate the probabilities for all the possibilities -- 2 white, 1 white/1 red, and then 2 red -- and see which method gives probabilities that actually add up to 1.

[sahilk47]

Hi Ron

I am still not able to understand your explanation pertaining to the case of one marble of each color. Why are we assuming that If Joan takes 2 random marbles from the jar implies sequential method? Why cant Joan take out two balls from the Jar simultaneously:

P(1 Red and 1 White) = [(8C1 * yC1)]/ [(8+y)*(7+y)]

Thank you.

[Chelsey Cooley]

It doesn't actually matter whether it's simultaneous or sequential - if it mattered, they'd have to make it clear in the problem. But in this case, the math comes out the same either way. Try to work it out both ways to see for yourself.

Hint: the denominator in your fraction isn't quite right - using the 'C' notation, it should be (8+y)C2. That's not quite the same as (8+y)*(7+y)...