Two cards are drawn from a standard deck of cards at the same time. Find:
a) Probability of drawing the King of hearts and a red card
b) Probability of drawing the King of hearts and a black card
Progress
a) I get a red card and a king hearts: P(R or K). There are 26 reds card out of 52 cards, so P(R) = 26/52 = 1/2. There are 4 kings out of 52 cards, so P(K) = 4/52 = 1/13. There is 1 card which is a red card and a king of hearts so
$$P(\text{H and K}) = \frac{1}{13}\times \frac12 = \frac{1}{26}$$
But I don't know if this is right.
Best Answer
One way is by a counting procedure. There are $\binom{52}{2}$ two-card hands. They are all equally likely.
We now count the favourables. There are $25$ hands that have the King of $\heartsuit$ and an additional red card. For there are $25$ red cards that are not the King of $\heartsuit$.
Thus the required probability is $\frac{25}{\binom{52}{2}}$.
For King of $\heartsuit$ and a black card, it's your turn.
Another way: Imagine drawing the cards one at a time (it makes no difference to the probability). We will be happy if (i) we draw the King of $\heartsuit$, and then another red card, or (ii) if we draw a red card other than the King of $\heartsuit$, and then the King of $\heartsuit$.
We find the probability of (1). The probability the first card is the King of $\heartsuit$ is $\frac{1}{52}$. Given that this happened, the probability the next card is red is $\frac{25}{51}$. So the probability of (i) is $\frac{1}{52}\cdot \frac{25}{51}$.
The probability of (ii) is the same. Add.