# Probability of drawing mutiple marbles.

#### billy1

##### New member
Hi, first post here...

Wasn't sure to put this in Basic Probability and Statistics or Advanced Probability and Statistics, figured I'd put it here in Advanced so it can get better help, if this belongs in Basic, mods feel free to move it.

There are 60 Marbles in a bag. There are X red marbles, Y blue marbles and Z yellow marbles.

X+Y+Z=60

If 7 Marbles are drawn from the bag (without replacement) what is the probability that at least 1 marble is red and 1 marble is blue. The order of drawing does not matter. The other 5 marbles can be red, blue or yellow, we only want to find out if at least 1 of the 7 marbles is red and 1 of the 7 marbles is blue.

My goal is to find a formula I can use to plug in different numbers for X, Y and Z. This is not for school, this is for a trading card game I play, I changed it to Marbles so you won’t get confused as the game does not use a standard 52 card deck, it uses 60 cards (hence marbles). Not sure if anyone here is familiar with trading card games.

The numbers I have been playing around with is X=4, Y=12, and Z=44.

This is what I did so far-

(56/60)*(55/59)*(54/58)*(53/57)*(52/56)*(51/55)*(50/54)

Above should be the 7 draws of the deck where a red marble (X=4) is not drawn.

Subtract the result I get from above from 1 gives me the probability of drawing 1 red marble off of 7 draws. Below is the formula for.

(1-(((Y+Z)-0)/60)*( (Y+Z)-1)/59)*( (Y+Z)-2)/58)*( (Y+Z)-3)/57)*( (Y+Z)-4)/56)*( (Y+Z)-5)/55)*( (Y+Z)-6)/54)))

Not even sure if this is even in the right direction to get what I am looking for, please help!

#### Plato

##### Well-known member
MHB Math Helper
Wasn't sure to put this in Basic Probability and Statistics or Advanced Probability and Statistics, figured I'd put it here in Advanced so it can get better help, if this belongs in Basic, mods feel free to move it.

There are 60 Marbles in a bag. There are X red marbles, Y blue marbles and Z yellow marbles.
X+Y+Z=60

If 7 Marbles are drawn from the bag (without replacement) what is the probability that at least 1 marble is red and 1 marble is blue. The order of drawing does not matter. The other 5 marbles can be red, blue or yellow, we only want to find out if at least 1 of the 7 marbles is red and 1 of the 7 marbles is blue.
What is the probability of drawing no red marbles or drawing no blue marbles.
Use inclusion/exclusion. Then find the complement.

#### Jameson

##### Administrator
Staff member
Hi billy1,

Welcome to MHB!

Whenever you see the words "at least" in probability a helpful way of calculating this is to realize that "at least 1 of something" is "1 - probability of none". Put more mathematically, $$\displaystyle P[X>0]=1-P[X=0]$$. Since this is discrete probability, $P[X>0]$ is the same thing as saying $P[X \ge 1]$. It doesn't matter which way you write it.

So as Plato wrote, you can do this problem by looking at "1-complement", or the opposite probability, for each event you wish to happen.

Jameson