Combinatorics and The Domino Effect

[kelly.yan]

I was reading on the topic of Combinatorics in the Word Translations MGMAT guide.

I had a hypothetical question I was hoping someone can help with. It's a revised version of an example question in the MGMAT Word Translation guide (pg 190). I changed the question a bit.

A miniature gumball machine contains 7 blue, 5 green and 4 red gumballs, which are identical except for their colors. If the machine dispenses three gumballs at random what is the probability that it dispenses 2 green gumballs and 1 blue gumball?

What would be the approach to this?

I'm getting 1/8. Is this correct??

The original question asked for the probability of the gumball machine dispensing 1 gumball of each colour. I wanted to see how dispensing 3 gumballs (2 of which are the same colour) affects the calculation.

Thanks!!

[rte.sushil]

I'm getting 1/8. Is this correct??

You are right if particular order is considered: 1/24
If you don't consider order of balls taken out: it will be 1/8

[kelly.yan]

Thanks for your help!

Any thoughts on how the fact that we're asking for 2 green gumballs and 1 blue affect the "order matters" probability?

I understand if we had asked for 1 of each color, probability would be 1/24 if order matters. But given that the first two gumballs must be green, is that probability still 1/24/ Shouldn't the probably be greater since we have two of the green gumballs? We could potentially have Green 1, Green 2 and Blue or Green 2 Green 1 and Blue.

Thoughts?

Thanks!!

[rte.sushil]

If order is considered, multiply by 3 if 2 Green and 1 blue, 3c2,
G G B , GBG, B GG

if three different, 3 factorial= 6
YBG, YGB, BYG, BGY, GBY, GYB

if all same, =1
BBB

usually by default order is not taken into consideration unless mentioned.

[kelly.yan]

I'm thinking through this, do you mean..
If order does not matter, then you multiply by 3 if we are looking for two green and one blue?

The logic here is that if order does not matter then all three permutations should be counted as part of the probability... GGB, GBG and BGG.

If order does matter, then the probability will be lower as we are only looking for GGB and not BBG or GBG.

I think where I'm confused is the situation where order does matter. In this case, we have two identical green gumballs and one blue gumball. Does that mean that we must count Green1, Green2 and Blue1 as a separate incident than Green2, Green1 and Blue1? Which means that probability of getting two green and one blue in a pool of 7 blue, 5 green and 4 red is 2/24. Or are these two the same and the actual probability is 1/24?

Thanks for clarifying!!

[tim]

the best way to deal with the issue of order is not to call it that at all. what you should do here is ask yourself what you calculated when you got 1/24 (presumably you started there before multiplying by 3). you got, for instance, the probability of green AND THEN green AND THEN blue. in the physical world, this is different than green blue green or blue green green. sure, the outcome is all the same, but the actual physical arrangement of atoms and molecules is different, so you tell yourself you found green then green then blue with 1/24 probability (call this scenario A). but then of course you would also get scenario B: green blue green with 1/24. or scenario C: blue green green with 1/24. since you want A or B or C, you add 1/24 + 1/24 + 1/24. this should make a lot more sense to you than if you just invoke some abstract question of whether "order matters". if you find yourself asking whether order matters without actually considering what happens in the scenario, you're just pushing symbols around on a page without understanding and run a high risk of getting the problem wrong..