Probability picking colored stones that match a series on a card. I'm inventing a board game that requires a person to randomly pick 4 colored gems out of a bag and have them match colored gems on a card. If I use ten each of four different colored gems, would probability of matching the gems on the cards change depending on the gems shown on the card? For example, the card might have all four gems the same color, there might be one of each color, or even two of one color, and two that are different colors.
Would odds increase if I added more of each colored gem (15 or 20), or decrease?
Thanks for any help you can give. Ultimately, I'm looking for greater probability, although hoping for under 5%. 
 A: To answer your question in general, if you have $4$ colors and $k$ gems of each color, then given a card of $4$ gems, the probability of picking a matching combination depends on the colors in the card.


*

*If all $4$ colors are the same: $\dfrac{\binom{4}{4}}{\binom{4k}{4}}$

*If $3$ colors are the same and $1$ color is different: $\dfrac{\binom{4}{3}\cdot\binom{4}{1}}{\binom{4k}{4}}$

*If $2$ colors are the same and $2$ other colors are the same: $\dfrac{\binom{4}{2}\cdot\binom{4}{2}}{\binom{4k}{4}}$

*If $2$ colors are the same and $2$ other colors are different: $\dfrac{\binom{4}{2}\cdot\binom{4}{1}\cdot\binom{4}{1}}{\binom{4k}{4}}$

*If all $4$ colors are different: $\dfrac{\binom{4}{1}\cdot\binom{4}{1}\cdot\binom{4}{1}\cdot\binom{4}{1}}{\binom{4k}{4}}$

The probabilities above will decrease as the value of $k$ increases.
But the ratio between them will remain constant regardless of the value of $k$.
This ratio is $1:16:36:96:256$ respectively.
A: Probability that all gems are green:
$$\frac{10}{40}\times\frac{9}{39}\times\frac{8}{38}\times\frac{7}{37}\approx0.23\%$$
Probability that one gem is green and the other three red:
$$\binom41\times\frac{10}{40}\times\frac{10}{39}\times\frac{9}{38}\times\frac{8}{37}\approx1.02\%$$
Probability that two gem are green and the other two, red:
$$\binom42\times\frac{10}{40}\times\frac{9}{39}\times\frac{10}{38}\times\frac{9}{37}\approx2.22\%$$
Probability that two gems are green, one red and one blue:
$$\frac{4!}{2!}\times\frac{10}{40}\times\frac{10}{39}\times\frac{10}{9}\times\frac{9}{37}\approx4.92\%$$
Probability that the four gems have different colours:
$$4!\times\frac{10}{40}\times\frac{10}{39}\times\frac{10}{38}\times\frac{10}{37}\approx10.94\%$$
Do you need also color-independent probabilities, like 'All gems are the same colour'?
About the effect of adding more gems of each color, that favours slightly the odds for repeated colours. For example, if there are $100$ gems each, the first probability (all gems green) turns to be:
$$\frac{100}{400}\times\frac{99}{399}\times\frac{98}{398}\times\frac{97}{397}\approx0.37\%$$
while the last (all gems different) becomes:
$$4!\times\frac{100}{400}\times\frac{100}{399}\times\frac{100}{398}\times\frac{100}{397}\approx9.52\%$$
A: There are two different effects here. One is that if you draw a gem of one colour, there is then one fewer gem of that colour in the bag, so the probability of drawing another one of the same colour is slightly less than drawing one of a different colour. This effect you can reduce to any desired degree by having more gems of each colour.
The second effect is quite different, and you can't reduce it by changing the number of gems. There are $4!=24$ different orders in which you can draw four gems of different colours, and only one order in which you can draw four gems of the same colour. Thus even if you completely eliminate the first effect (by having infinitely many gems or by putting the gems back into the bag before drawing the next one), the probability of drawing gems of four different colours will always be at least $24$ times the probability of drawing four gems of the same colour.
You can also think about it like this: When you draw the first gem, what's your chance of drawing a gem that you need? If you're holding a four-colour card, you can use whatever colour you draw; if you're holding a monochromatic card, you only have a $1$ in $4$ chance to draw that one colour you need. For the second gem, it's again $1$ in $4$ for the monochromatic card, whereas still $3$ out of $4$ colours are allowed for the multi-coloured card. Doing the same thing for the third and fourth gem and multiplying the results yields the same factor $24$ between the two probabilities.
If you want the cards to have similar probabilities, you'd need to have more gems on the multi-coloured ones and fewer gems on the monochromatic ones.
