# Probability of drawing marbles

#### kory

##### Junior Member
I have 5 marbles in a bag: A red, orange, blue, purple, green.
I take one marble out, then I put it back. I shake the bag and then draw another marble. What is the chances of drawing the same color?
What is the probability of getting at least one blue marble in your two draws ?

To draw the same color would I just multiply 5 x 4 x 3 x 2 x 1 ?...

#### pka

##### Elite Member
I have 5 marbles in a bag: A red, orange, blue, purple, green.
I take one marble out, then I put it back. I shake the bag and then draw another marble. What is the chances of drawing the same color?
What is the probability of getting at least one blue marble in your two draws ?
For part 1. The probability of two reds is: $$\mathcal{P}(R_1R_2)=\left(\dfrac{1}{5}\right)^2=\dfrac{1}{25}$$
But there are five colours. So, what are the chances of drawing two of the same color?

#### JeffM

##### Elite Member
I have 5 marbles in a bag: A red, orange, blue, purple, green.
I take one marble out, then I put it back. I shake the bag and then draw another marble. What is the chances of drawing the same color?
What is the probability of getting at least one blue marble in your two draws ?

To draw the same color would I just multiply 5 x 4 x 3 x 2 x 1 ?...
A probability is ALWAYS a number in the interval [0, 1]. So your answer cannot possible be right.

#### kory

##### Junior Member
For part 1. The probability of two reds is: $$\mathcal{P}(R_1R_2)=\left(\dfrac{1}{5}\right)^2=\dfrac{1}{25}$$
But there are five colours. So, what are the chances of drawing two of the same color?
I got $$\displaystyle \frac {1!}{2!}$$ /$$\displaystyle \frac {5!}{3!2!}$$ = $$\displaystyle \frac 1{20}$$

#### pka

##### Elite Member
I got $$\displaystyle \frac {1!}{2!}$$ /$$\displaystyle \frac {5!}{3!2!}$$ = $$\displaystyle \frac 1{20}$$
$$\begin{array}{*{20}{c}} {(R,R)}&{(R,O)}&{(R,B)}&{(R,P)}&{(R,G)} \\ {(O,R)}&{(O,O)}&{(O,B)}&{(O,P)}&{(O,G)} \\ {(B,R)}&{(B,O)}&{(B,B)}&{(B,P)}&{(B,G)} \\ {(P,R)}&{(P,O)}&{(P,B)}&{(P,P)}&{(P,G)} \\ {(G,R)}&{(G,O)}&{(G,B)}&{(G,P)}&{(G,G)} \end{array}$$
Above is the outcome space. What is the probability of selecting the same colour?

#### JeffM

##### Elite Member
Whatever color you draw first, how many ways can you draw the next marble to be the same color? How many distinct ways can you draw the second marble.

By the way, you are essentially guessing. What is the significance of 1!/2!. Why 5!/3!? You should have reasons for why you choose your numerators and denominators.

#### kory

##### Junior Member
$$\begin{array}{*{20}{c}} {(R,R)}&{(R,O)}&{(R,B)}&{(R,P)}&{(R,G)} \\ {(O,R)}&{(O,O)}&{(O,B)}&{(O,P)}&{(O,G)} \\ {(B,R)}&{(B,O)}&{(B,B)}&{(B,P)}&{(B,G)} \\ {(P,R)}&{(P,O)}&{(P,B)}&{(P,P)}&{(P,G)} \\ {(G,R)}&{(G,O)}&{(G,B)}&{(G,P)}&{(G,G)} \end{array}$$
Above is the outcome space. What is the probability of selecting the same colour?
$$\displaystyle \frac {1}{25}$$ + $$\displaystyle \frac {1}{25}$$ = .08 ?
1 out of 25 for the first attempt and another 1 out of 25 for the second attempt

#### pka

##### Elite Member
$$\displaystyle \frac {1}{25}$$ + $$\displaystyle \frac {1}{25}$$ = .08 ?
1 out of 25 for the first attempt and another 1 out of 25 for the second attempt
There are twenty-five pairs in the outcome space.
In how many are there two of the same colours?
What is the probability of pair with same two colours?

#### JeffM

##### Elite Member
STOP GUESSING and think.

What is 1/25 supposed to represent?

How many ways can your pick a different color on the second draw given whatever color you drew on the first draw?

How many ways can you draw a marble the second time.

So the probability of drawing a marble of a different color is what?

Therefore the probability of drawing a marble of the same color is what?

#### kory

##### Junior Member
There are twenty-five pairs in the outcome space.
In how many are there two of the same colours?
What is the probability of pair with same two colours?
There are 5 with the same color...{RR,OO,PP,BB,GG}

my notes say "number of outcomes in the event / number of outcomes in sample space:"
thats 2/25 which = 8%
2 for the 2 marbles that i take out of the bag(event) , and 25 for the sample space....unless there is another formula then im not sure what you guys are asking.

#### pka

##### Elite Member
Well, I give-up. kory you need serious help from a live tutor.

#### JeffM

##### Elite Member
"Number of outcomes in the event" is meaningless. I do not know what you heard, but it was not what your teacher said.

Number of possible favorable outcomes / number of all possible outcomes (assuming each each possible outcome is equally likely)

So you count out 5 favorable outcomes (both same color). You actually list them RR, OO, PP, BB, GG. That is five outcomes with the same color.

You see that the number of possible outcomes is 25.

How does 5/25 become 2/25?

EDIT: Please notice the proviso that counting numbers works only if each outcome is equally probable.

EDIT 2: I have figured out what "number of outcomes in the event" means. "Favorable event" is same color. How many outcomes give the favorable event? I do not love the language, but I can see how it might be used in a context of discussing "favorable event." I'd prefer to say "ratio of number of possible favorable events to the number of all possible events if each possible event is equally likely" or "ratio of possible favorable outcomes to all possible outcomes if each possible outcome is equally likely." I think being inconsistent about the terms "event" and "outcome" is bound to lead to confusion.

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#### kory

##### Junior Member
I thought the event was the number of marbles drawn. If its favorable outcome then I guess it would be 5 for the 5 matching colors.

I dont mean to drive you crazy but my school is shut down because of the bug so this is the only place I can come to for help...

#### JeffM

##### Elite Member
I thought the event was the number of marbles drawn. If its favorable outcome then I guess it would be 5 for the 5 matching colors.

I dont mean to drive you crazy but my school is shut down because of the bug so this is the only place I can come to for help...
Yes, the probability that the second ball will match the first ball is 5/25 = 20%.

"Event" here is a sequence of colors. You want to find the probability of one of the favorable events, here that the first and second ball match in color. To use the language of events or outcomes, you must figure out anew in each problem what "event" or "outcome" and "favorable" mean in the context of that problem.

I suspect that if the problem had asked you to figure out the probability that both the first and second balls were red, you would have got 1/25. But here we are asking for the probability that they are both red, both orange, both blue, both purple, or both green. They are mutually exclusive so we just add them up,

(1/25) + (1/25) + (1/25) + (1/25) + (1/25) = 5/25

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#### kory

##### Junior Member
Yes, the probability that the second ball will match the first ball is 5/25 = 20%.

"Event" here is a sequence of colors. You want to find the probability of one of the favorable events, here that the first and second ball match in color. To use the language of events or outcomes, you must figure out anew in each problem what "event" or "outcome" and "favorable" mean in the context of that problem.

I suspect that if the problem had asked you to figure out the probability that both the first and second balls were red, you would have got 1/25. But here we are asking for the probability that they are both red, both orange, both blue, both purple, or both green. They are mutually exclusive so we just add them up,

(1/25) + (1/25) + (1/25) + (1/25) + (1/25) = 5/25
Thank you so much, I really appreciate the help. I have an exam comming in a few weeks and i'm trying to prepare.