A miniature gumball machine contains 7 blue, 5 green

[noravoningersleben]

MGMAT World Translations Strategy Guide, 3rd ed
Chapter 5 - Probability Strategy
p. 98

Wording of problem:

"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 one gumball of each color?"

Explanation (relevant part for my question):

"In general, when you have a symmetrical problem with multiple equivalent cases, calculate the probability of one case (often by using the domino-effect rule). Then multiply by the number of cases. Use combinatorics to calculate the number of cases, if necessary. Remember that when you use a symmetry argument, the situation must truly be symmetrical. In the case above, if you swapped the order of "red" and "green" emerging from the gumball machine, nothing would change about the problem. As a result, we can use symmetry to simplify the problem."

My question:

In spite of the explanation given in the book (and quoted above), I still don't really know what "symmetry" means when applied to probability. Does symmetry mean that each outcome has an equal chance of happening? Could you maybe give an example of a case that is not symmetrical? Thanks for your help!!

[shyamprasadrao]

let me know if the answer for the problem is 1/24. If I am correct, I can explain. Otherwise I might confuse you.

[nehag84]

What is the answer?

[noravoningersleben]

Sorry for the delay. Here is the complete answer from the book:

"Consider one specific case: blue first, then gren, then red. By the domino-effect rule, the probability of this case is 7 blue/16 total * 5 green/15 total * 4 red/14 total = 7/16 * 5/15 * 4/14 = 1/24.

Now consider another case: green firstm then red, then blue. The probability of this case is 5 green/16 total * 4 red/15 total * 7 blue/14 total = 5/16 * 4/15 * 7/14 = 1/24. Notice that all we have done is swap around the numerators. We get the same final probability! This is no accident; the order in which the balls come out does not matter.

Because the three desired gumballs can come out in any order, there are 3! = 6 different cases. All of these cases must have the same probability. Therefore, the overall probability is 6 * 1/24 = 1/4.

In general, when you have a symmetrical problem, with multiple equivalent cases, calculate the probability of one case (often by using the domino-effect rule). Then multiply by the number of cases. Use combinatorics to calculate the number of cases, if necessary.

Remember that when you use a symmetry argument, the situation must truly be symmetrical. In the case above, if you swapped the order of "red" and "green" emerging from the gumball machine, nothing would change about the problem. As a result, we can use symmetry to simplify the computation."

[sd]

Guys, actually this can be made much simpler...you dont have to bring order of the gum balls into the picture here at all. All we need is choosing (combinations).

Number of ways of chosing 3 gum balls out of 16 total gum balls = 16C3.
Number of ways of chosing 1 blue, 1 green and 1 red ball = 7C1 X 5C1 X 4C1

So probability = 7C1 X 5C1 X 4C1 / 16C3
= 7*5*4*3*2 / 16*15*14
= 1/4

[noravoningersleben]

Hi SD,

Thanks - that works!

I'm still not sure, though, what symmetry is... oh well...

Thanks everybody for your responses.

[Ben Ku]

Hi,

By Symmetry, the Strategy Guide is talking about possibilities that have the same probability.

For example, the probability of BGR is calculated as (7/16)(5/15)(4/14) = 1/24.
The probability of GRB is (5/16)(4/15)(7/14) = 1/24.

Notice that even though different fractions are being multiplied, the end probabilities are the same. So the conclusion is that no matter what order we pick one of three colors, the probability of each will be 1/24. This is what is meant by symmetry in this context.

Hope that helps!

[jp.jprasanna]

Hi - Could you please let me know what's wrong with my method below ->

Nos of ways to select 3 gumball from a total of 16 gumball is 560

probability that it dispenses one gumball of each color =
1 - probability that it dispenses same colour

1 - (ALL 3 BLUE or ALL GREEN or ALL RED / TOTAL )
1 - (7c3 + 5c3 + 4c3 / 560) which is not equal to 1/4


what am i doing wrong here?

[RonPurewal]

probability that it dispenses one gumball of each color =
1 - probability that it dispenses same colour

this is false.

if you do 1 minus the probability that everything is the same color, then you get ALL of the possibilities for which "everything is the same color" is false.
these possibilities include the cases in which all the colors are different (= the cases that you actually want here), but, unfortunately, they also include the cases in which exactly two of the three colors are the same (you don't want these).