Given a deck of 40 cards, with 4 suits of 10 cards each and cards numbered from 1 to 10, we draw 4 cards. Let the random variable N be the number of different suits. I am asked to compute $P(N=2)$.
The solution provided by using Laplace rule is:
Number of possible draws: ${40 \choose 4}=91390$
Number of positive outcomes: $4{10 \choose 3}*3*10+{4 \choose 2}{10 \choose 2}{10 \choose 2}=26550$
So $P(N=2)\approx0.29$.
The reasoning for this I understand. But I appraoched the problem differently and the result I get is different and therefore wrong. Here it is:
I first computed the probability of drawing, say, hearts, hearts, spades and spades, which would be $P(HHSS) = \frac{10}{40}\frac{9}{39}\frac{10}{38}\frac{9}{37}$. We want to take different orderings into account, so we multiply by ${4 \choose 2}$. And also we don't only want hearts and spade, but all suit combinations, so I multiply by ${4 \choose 2}$ again.
This is for having 2 of each suit, but we can also have 1 and 3 of each, so similarly:
$P(HHHS)=\frac{10}{40}\frac{9}{39}\frac{8}{38}\frac{10}{37}$. We also take ordering into account so multiply by ${4 \choose 1}$ and same as above, I account for the different suit combinations with ${4 \choose 2}$. The final expression is:
$P(N=2)={4 \choose 2}{4 \choose 2}\frac{10}{40}\frac{9}{39}\frac{10}{38}\frac{9}{37}+{4 \choose 2}{4 \choose 1}\frac{10}{40}\frac{9}{39}\frac{8}{38}\frac{10}{37}$
but this results in a number close 0.21.
Could someone please help me figure out what is wrong?
Thanks in advance
The reason my calculation was wrong is due to the fact that I was only counting the number of ways in which I can combine HHHS (3 hearts 1 spade) by changing the position of S with the term ${4 \choose 1}$ but I also need the number of ways in which SSSH (3 spades and 1 heart) can occur, as pointed out. So by multiplying the second term by 2 (due to summetry) the result is now correct.