# Thread: balls chosen at random from urn

1. ## balls chosen at random from urn

Q: An urn contains two blue balls B_1 and B_2 and three white ball W_1, W_2, W_3. One ball is drawn, its colour is recorded and its replaced in the urn. Then another ball is drawn and its colour is recorded.

a)Let B_1W_2 denote the outcome the first ball drawn is B_1 and the second is W_2. List all 25 outcomes.

b)Consider the event that only white balls are drawn. List the outcomes and the probability.

to make this faster im just going to type B1 or W3

a) S = {B1B1,B1B2,B2B2,B2B1,W1W1,W1W2,W1W3,W2W2,W2W3,W3W3 ,W2W1,W3W2,W3W1,B1W1,B1W2,B1W3,B2W1,B2W2,B2W3,W1B1 ,W2B1,W3B1,W2B1,W2B2,W2B3,W3B1,W3B2,W3B3 [this is 28 outcomes so far but it says there are only 25]

b) I know this is $\displaystyle P(E) = \frac{N(E)}{N(S)}$ but I need to figure out what went wrong in a before I can answer this

2. ## Re: balls chosen at random from urn

each draw is from 5 unique objects.

Of course there are 5x5 ways to uniquely select pairs from the 5.

Treat each ball as a base 5 digit. Your possibilities are all 2 digit base 5 numbers.

I see B3's in your list. You only have B1 and B2.

for (b) you only have 3 unique objects now. So there are 3x3=9 combinations. To list them list all 2 digit base 3 numbers.

the probability is $\dfrac 9 {25}$. 9 possible combinations out of 25.

3. ## Re: balls chosen at random from urn

ah yeah, I think I got in a pattern there and included a B3 out of the repetition

thanks

4. ## Re: balls chosen at random from urn

Hello, Jonroberts74!

An urn contains two blue balls B_1 and B_2 and three white ball W_1, W_2, W_3.
One ball is drawn, its colour is recorded and its replaced in the urn.
Then another ball is drawn and its colour is recorded.

(a) Let B_1W_2 denote: the first ball drawn is B_1 and the second is W_2.
. . List all 25 outcomes.

$\displaystyle \begin{array}{c|cccccc} & B_1&B_2&W_1&W_2&W_3 \\ \hline B_1 & B_1B_1 & B_1B_2 & B_1W_1 & B_1W_2 & B_1W_3 \\ B_2 & B_2B_1 & B_2B_2 & B_2W_1 & B_2W_2 & B_2W_3 \\ W_1 & W_1B_1 & W_1B_2 & W_1W_1 & W_1W_2 & W_1W_3 \\ W_2 & W_2B_1 & W_2B_2 & W_2W_1 & W_2W_2 & W_2W_3 \\ W_3 & W_3B_1 & W_3B_2 & W_3W_1 & W_3W_2 & W_3W_3 \end{array}$

(b) Consider the event that only white balls are drawn.
List the outcomes and the probability.

$\displaystyle \begin{array}{c|ccc} & W_1 & W_2 & W_3 \\ \hline W_1 & W_1W_1 & W_1W_2 & W_1W_3 \\ W_2 & W_2W_1 & W_2W_2 & W_2W_3 \\ W_3 & W_3W_1 & W_3W_2 & W_3W_3 \end{array}$

$\displaystyle P(\text{both W}) \:=\:\frac{9}{25}$