Consider a scenario where Ace is considered as 1 (a number card) and the face cards (King, Queen, Jack) are not number cards. We have to find the probability of 3 cards drawn consecutively (without replacement) being the same number. My friend solved it using the following:
$$P = \frac{^4C_1\cdot ^{10}C_1\cdot ^3C_1\cdot ^2C_1 }{^{52}C_3} = \frac{12}{1105}$$
where we choose any one out of the 4 types, and then we choose a number card, and then we choose the same number card in the other types twice.
I did the same problem like the following:
$$P = \frac{10}{13}\cdot \frac{3}{51}\cdot \frac{2}{50} = \frac{2}{1105}$$
where $\frac{10}{13}$ is the probability of choosing a number card first, $\frac{3}{51}$ is the probability of choosing the same number card out of the rest of 51 cards and $\frac{2}{50}$ is the probability of choosing the same number card out of the rest of 50 cards.
Could you please explain as to why we are getting different answers and which one is correct (I guess it has to do something with $6$ multiplication error in either one of the answers arising from problems in permutations (as $6$ = $3!$) of the types of cards)?
Best Answer
The first method is incorrect as the numerator distinguishes between hands with the same cards but different order, but the denominator does not. Thus the first method would say that the probability of drawing $\{2\heartsuit, 2\spadesuit, 2\clubsuit\}$ was $\frac 6{\binom {52}3}$ where it is clearly $\frac 1{\binom {52}3}$ (as all specified hands are equally probable). In other words, you were correct in your intuition that the factor $6=3!$ arises from the possible permutations of a three card hand.
The second method is correct. Another way to see it is to note that there are $10$ possible number values and $\binom 43$ ways to choose a three card hand with those number values, so the probability is $$\frac {10\times \binom 43}{\binom {52}3}=\frac 2{1105}$$