Question

Bag $A$ contains $6$ Green and $8$ Red balls and bag $B$ contains $9$ Green and $5$ Red balls. A card is drawn at random from a well shuffled pack of $52$ playing cards. If it is a spade, two balls are drawn at random from bag $A$, otherwise two balls are drawn at random from bag $B$. If the two balls drawn are found to be of the same colour, then the probability that they are drawn from bag $A$ is

• A. $\frac{43}{181}$
• B. $\frac{1}{4}$
• C. $\frac{48}{131}$
• D. $\frac{43}{138}$

Answer 1

Answer: (A)

$\frac{43}{181}$

Answer 2

According to given informations, the required probability
$=\frac{\frac{1}{4}\left(\frac{{ }^{6} C_{2}+{ }^{8} C_{2}}{{ }^{14} C_{2}}\right)}{\frac{1}{4}\left(\frac{{ }^{6} C_{2}+{ }^{8} C_{2}}{{ }^{14} C_{2}}\right)+\frac{3}{4}\left(\frac{{ }^{9} C_{2}+{ }^{5} C_{2}}{{ }^{14} C_{2}}\right)}$
$=\frac{(6 \times 5)+(8 \times 7)}{[(6 \times 5)+(8 \times 7)]+3[(9 \times 8)+(5 \times 4)]}$
$=\frac{30+56}{(30+56)+3(72+20)}=\frac{86}{86+276}$
$=\frac{86}{362}=\frac{43}{181}$