RI Probability Tutorial Q8

In a probability experiment, three containers have the following contents.

A jar contains \({2}\) white dice and \({3}\) black dice.

A white box contains \({5}\) red balls and \({3}\) green balls.

A black box contains \({4}\) red balls and \({3}\) green balls.

One die is taken at random from the jar. If the die is white, two balls are taken from the white box, at random and without replacement. If the die is black, two balls are taken from the black box, at random and without replacement.

Events \({W}\) and \({M}\) are defined as follows.

\({W: }\) A white die is taken from the jar.

\({ M: }\) One red ball and one green ball are obtained.

Show that \(\mathrm{P}\left( M\left| W \right. \right)=\frac{15}{28}\).[2]

Find, giving each of your answers as an exact fraction in its lowest terms,

1. \({P(M \cap W)}\),[1]
2. \(\mathrm{P}\left( W\left| M \right. \right)\),[5]
3. \({P(W \cup M)}\).[3]

All the dice and balls are now placed in a single container, and four objects are taken at random, each object being replaced before the next one is taken. Find the probability that one object of each colour is obtained.