So we draw cards from a deck and we want to know the probability of taking a king of hearts.
The first thinking process:
We draw 2 cards together and we do not return them: $$P(A) = \frac{1\times \binom{51}{1}}{\binom{52}{2}}= \frac{2}{52}$$
The second thinking process:
We draw 2 cards one by one:
$$P(A)=\frac{1\times 51 + 1\times 51}{52\times 51} = \frac{2}{52} $$
I know that this is the same thing, but in classes we said that the thinking process is different between the upper two procedures. Can somebody explain for both of them?
Order of selection matters in the second method but not the first.
In the first method, what matters is which cards are selected. The denominator counts the number of ways we can select two cards from the deck without regard to order, which is $\binom{52}{2}$. The numerator counts the number of ways we can draw a king of hearts and one of the other $51$ cards in the deck without regard to the order of selection, which is $\binom{1}{1}\binom{51}{1}$ since we select the only king of hearts and one of the other $51$ cards in the deck.
In the second method, we do consider the order of selection. There are $52$ ways to draw the first card and $51$ ways to draw the second card from among the remaining cards, giving $52 \cdot 51$ ordered selections of two cards. If we draw the king of hearts, it must either be drawn first and another card must be drawn next ($1 \cdot 51$) or another card must be drawn first followed by the king of hearts ($51 \cdot 1$). Since these two possibilities are mutually exclusive, we add them to find the number of ways of drawing a king of hearts when two hearts are drawn from the deck.