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.

Answer 1

Let E₁, E₂ , E₃ ,E₄ and A be event defined as 

E₁ = the lost card is a spade card. 

E₂ = the lost card is a heart card. 

E₃ = the lost card is a club card. 

E₄ = the lost card is diamond card. 

and A = Drawing three spade cards from the remaining cards.

P(E₁) = P(E₂) = P(E₃) = P(E₄) =  1/4

P(A/E₁) = ¹²C₃/⁵¹C₃ = 220/20825, P(A/E₂) = ¹³C₃/⁵¹C₃ = 286/20825

P(A/E₃) = ¹³C₃/⁵¹C₃ = 286/20825 P(A/E₄) = ¹³C₃/⁵¹C₃ = 286/20825

By using Baye’s theorem, we have

P(E₁/A) = (P(E₁)P(A/E₁))/(P(E₁)P(A/E₁) + P(E₂)P(A/E₂) + P(E₃)P(A/E₃) + P(E₄)P(A/E₄))