Question

$3$ cards are given, one of them is red on both sides, one is blue on both sides & one is blue on one side red on the other side. One of them is chosen randomly & put on the table. It shows red colour on the upper side. The chance of the other side of the card being red is:

Answer 1

Let $E_1,E_2,E_3$ be the event of choosing a card red on both sides , card blue on both sides and blue on side & red on other side

Let A be the event that the drawn card shows red on upper side .

then , $P\left(\frac{A}{E_1}\right)=\frac{2}{2}=1$

$P\left(\frac{A}{E_2}\right)=\frac{0}{2}=0$

$P\left(\frac{A}{E_3}\right)=\frac{1}{2}$

$P\left(E_1\right)=P\left(E_2\right)=P\left(E_3\right)=\frac{1}{3}$

$P\left(\frac{E_1}{A}\right)=\frac{P\left(E_1\right)\times P\left(\frac{A}{E_1}\right)}{P\left(E_1\right)\times P\left(\frac{A}{E_1}\right)+P\left(E_2\right)\times P\left(\frac{A}{E_2}\right)+P\left(E_3\right)\times P\left(\frac{A}{E_3}\right)}$

$=\frac{\frac{1}{3}\times 1}{\frac{1}{3}\times 1+\frac{1}{3}\times 0+\frac{1}{3}\times \frac{1}{2}}$

$=\frac{\frac{2}{6}}{\frac{2}{6}+\frac{1}{6}}$

$=\frac{2}{3}$