A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.

#### Solution

Let E_{1}_{, }E_{2}_{, }E_{3}_{, }E_{4} and A be the event defined as below:

E_{1} = the missing card is a heart card

E_{2}_{ = }the missing card is a spade card

E_{3}_{ = }the missing card is a club card

E_{4}_{ = }the missing card is a diamond card

A = drawing three spade cards from the remaining cards

Now, we have the following:

`P(E_1)=13/52=1/4`

`P(E_2)=13/52=1/4`

`P(E_3)=13/52=1/4`

`P(E_4)=13/52=1/4`

`P(A/E_1)=(""^13C_3)/(""^51C_3)`

`P(A/E_2)=(""^12C_3)/(""^51C_3)`

`P(A/E_3)=(""^13C_3)/(""^51C_3)`

`P(A/E_4)=(""^13C_3)/(""^51C_3)`

By Bayes Theorem, we have:

Required probability =`P(E_2/A)`

`=(P(E_1)P(A/E_2))/(P(A/E_1)E_1+P(A/E_2)E_2+P(A/E_3)E_3+P(A/E_4)E_4)`

`=(1/4xx(""^13C_3)/(""^51C_3))/((""^13C_3)/(""^51C_3)xx1/4+(""^12C_3)/(""^51C_3)xx1/4+(""^13C_3)/(""^51C_3)xx1/4+(""^13C_3)/(""^51C_3)xx1/4)`

`=(""^12C_3)/(""^12C_3+3xx""^13C_3)`

`=220/(220+286xx3)=110/539`