You have 30 cards 10 red 10 green and 10 blue The cards of e

You have 30 cards: 10 red, 10 green, and 10 blue. The cards of each color are labeled 1, 2, 3, ..., 10. You randomly draw three cards. What is the probability that all three card values are the same given that at least one of the card values is repeated?

Solution

P(all three values are same) = 10/ ³⁰C₃

P(at least one of the card values is repeated)

= P(all three card is repeated) + P(2 cards are repeated)

= 10/ ³⁰C₃+ 3*9*10 / ³⁰C₃

= 10+270/ ³⁰C₃

= 280 / ³⁰C₃

P(all three values are same | at least one of the card values is repeated )

= P(all three values are same, at least one of the card values is repeated ) / P(at least one of the card values is repeated)

= P(all three values are same) / P(at least one of the card values is repeated)

= [ 10/ ³⁰C₃ ] / [ 280 / ³⁰C₃ ]

10 / 280

= 1/ 28 = 0.0357