# Drawing balls from a box based on die toss

Two boxes, the first contains 3 red & 2 white balls. The second box contains 4 red & 7 white balls.

A fair die is tossed. If the outcome is even, a ball is drawn from the first box, while a ball is drawn from the second box if the outcome is odd.

Find the probability that a randomly chosen ball will be white and it comes from the first box.

Attempt at solution:

P(drawing from first box)= 3/6= 1/2, P(drawing a white ball)=9/16, P(White ball ∩ first box)= P(White ball | first box)xP(First box) =(2/5)(1/2)=1/5,

My friend says that the required probability is not P( White ball ∩ first box) but instead it's P(White ball | first box), which is it?

• It is just a matter of reading the question correctly ("...white and it comes from the first box), and you did that. Your answer is correct. Your friend makes it: "...white if it comes from the first box". – drhab Oct 17 '15 at 18:43
• Your interpretation is correct. – true blue anil Oct 17 '15 at 19:16

For questions about conditional probability, usually the word given will be in the phrase somewhere.

• Find the probability that a randomly chosen ball will be white and it comes from the first box.
• Find the probability that a randomly chosen ball will come from the first box and be white.

both represent $P(\text{White}\cap \text{First box})$

• Find the probability that a randomly chosen white ball comes from the first box.
• Find the probability that a randomly chosen ball comes from the first box given that it is white.

represents $P(\text{First box}~|~\text{White})$

• Find the probability that a randomly chosen ball from the first box is white.
• Find the probability that a randomly chosen ball is white given that it comes from the first box.

represents $P(\text{White}~|~\text{First box})$

I agree with your interpretation and that the probability being searched for is indeed $P(\text{White}\cap \text{First box}) = \frac{1}{5}$

The probability that the ball gets chosen from the first box is $\frac{1}{2}$, and the proability that a ball chosen from the first box is white is $\frac{2}{5}$.

The probability that a ball gets chosen from the first box and that it is white is $\frac{1}{2} \times \frac{2}{5} = \frac{1}{5}$.