Answer to Question #160296 in Statistics and Probability for VayaMan

Question

8. Two identical bags each contain 12 discs, which are identical except for a) bag is selected at random and a disc is selected from it. Draw a tree diagram illustrating this situation and calculate the probability that the disc drawn will be red. b) The disc selected is returned to the same bag, along with another two the same colour, and another disc is chosen from that bag. Find the probability that i) it is the same colour as the first disc drawn ii) bag A was used, given that two discs the same colour have been chosen. 4.5 Relationships between events colour. Bag A contains 6 red and 6 blue discs. Bag B contains 8 red and 4 blue discs.

Answer 1

Bag A contains 6 red and 6 blue disks. Bag B contains 8 red and 4 blue disks.

a)

$$P(Red)=P(A\cap Red)+P(B\cap Red)$$

$$=1/4+1/3=7/12$$

b)

i)

$$P(same\ colour)$$

$$=P(A\cap Red)\cdot P(Red|A)+P(B\cap Red)\cdot P(Red|B)$$

$$+P(A\cap Blue)\cdot P(Blue|A)+P(B\cap Blue)\cdot P(Blue|B)$$

$$=\dfrac{1}{4}(\dfrac{1}{2})+\dfrac{1}{3}(\dfrac{2}{3})+\dfrac{1}{4}(\dfrac{1}{2})+\dfrac{1}{6}(\dfrac{1}{3})=\dfrac{19}{36}$$

ii)

$$P(A|same\ colour)$$

$$=\dfrac{P(A\cap Red)\cdot P(Red|A)+P(A\cap Blue)\cdot P(Blue|A)}{P(same\ colour)}$$

$$=\dfrac{\dfrac{1}{4}(\dfrac{1}{2})+\dfrac{1}{4}(\dfrac{1}{2})}{\dfrac{19}{36}}=\dfrac{9}{19}$$