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Question
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.
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Solution
Let E1, E2, E3, E4 and A be the event defined as below:
E1 = the missing card is a heart card
E2 = the missing card is a spade card
E3 = the missing card is a club card
E4 = 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`
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