Question

A card from pack of 52 cards is lost. From the remaining card of the pack, one card is drawn and is found to be heart, find the probability of missing card to be (I) heart (II) club.

Answer 1

Let$E_1$be the event that the lost card is a diamond. Given that there are $13$ diamonds in the deck,$P\left(E_2\right)=\frac{13}{52}=\frac{1}{4}$

Let $E_2$ be the event that the lost card is not a diamond. $P\left(E_2\right)=1-\frac{13}{52}=\frac{39}{52}=\frac{3}{4}$

Let A be the event that the two cards drawn are found to be both diamonds.

We need to calculate P ($2$ cards are both diamond w the lost card being a diamond) next.

Given $13$ diamond cards, if one is lost, then we have $12$ remaining diamonds out of $51$ total cards now.

The two cards that are both diamond can be drawn in ${}^{12}{C}_2=\frac{12\times 11}{1\times 2}=\frac{132}{2}$= $66$ ways.

Likewise, the two diamonds can be drawn from the pack of $51$ remaining cards in ${}^{51}{C}_2=\frac{51\times 50}{1\times 2}=\frac{2550}{2}$= 1275 ways.

Therefore, the P ($2$ cards drawn are diamond given one is lost) = $P\left(A|E_1\right)\frac{66}{1275}$

Now, consider the event where the two cards drawn are both diamonds but the lost card is not a diamond.

The two cards that are both diamond can be drawn in ${}^{13}{C}_2=\frac{13\times 12}{1\times 2}=\frac{156}{2}$= 78ways.

Likewise, the two diamonds can be drawn from the pack of 51 remaining cards in ${}^{51}{C}_2=\frac{51\times 50}{1\times 2}=\frac{2550}{2}$ = 1275 ways.

Therefore, the P (2 cards drwan are diamond given one card which is not a diamond is lost) = $P\left(A|E_1\right)\frac{78}{1275}$

We need to calcuate the probability that the lost card is a diamond. $P\left(E_1|A\right)$.

We can use Baye's theorem, according to which $P\left(E_1|A\right)=\frac{\frac{66}{1275}.\frac{1}{4}}{\frac{66}{1275}.\frac{1}{4}+\frac{78}{1275}.\frac{3}{4}}=\frac{66}{66+78\times 3}=\frac{66}{300}=\frac{11}{50}$

$P\left(E_1|A\right)=\frac{\frac{66}{1275}.\frac{1}{4}}{\frac{66}{1275}.\frac{1}{4}+\frac{78}{1275}.\frac{3}{4}}=\frac{66}{66+78\times 3}=\frac{66}{300}=\frac{11}{50}$