Three cards are drawn at random from a pack of well shuffled $52$ cards. Find the probability that all the $3$ cards are of the same suit.

My two attempts:

  1. $\displaystyle\frac{ {13 \choose 3} }{ 52 } = \frac{ 11 }{ 850 } $

  2. $\displaystyle\frac{ 13 \cdot 13 \cdot 13 }{52 \cdot 52 \cdot 52 } = \frac1{64} $

Either way the answer is wrong as given in the book, which is $\frac{22}{425}$ .

What am I doing wrong?

  • $\begingroup$ Why do you think the official answer is wrong? Hint: the first can be anything. The probability that the second has the same suit is then $\frac {12}{51}$. The probability that the third also matches is then... $\endgroup$ – lulu Dec 30 '18 at 12:24

Your first method is almost correct. Your error is that you only considered one suit, but there are four suits in the deck. Therefore you have to multiply your solution by 4, which gives the correct solution.


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