Probability of specific card being in the deck vs your players hand?

Question

If there are 7 cards left in a deck of cards, and 5 cards in the other players hand and you've seen all the cards except for the following:

Spades: ♠ 10 K

Hearts:♥ 10 Q K A

Clubs: ♣ 10 J Q K A

Diamonds: ♦ A

1. What is the probability that the Ace of Diamonds is in the deck?
2. What is the probability that the other player's hand contains at most 1 spade?
3. What is the probability that the other player's hand contains at most 1 spade, given that the ace of diamonds is in the deck?

My first thought for number 1 is simply P(A) = 7/12, but I have a feeling that is wrong.

For number 2, wouldn't the answer be 762/792?

Since $\frac{\binom{2}{2}\binom{10}{3}}{\binom{12}{5}}$ represents the number of hands where both spades are drawn,

P(B) = 1- $\frac{\binom{2}{2}\binom{10}{3}}{\binom{12}{5}}$ = $\frac{762}{792}$ where A is the event that at most one spade is drawn.

For number 3, I know in order to compute P(B|A), you need to know the intersection of A $\cap$ B, which I am confused about how to compute because they are aren't disjoint.

How do I go about solving these?

Answer 1

To justify your answer for Question 1: Let's count the number of ways the Ace of Diamonds could be in the deck. There would have to be $6$ other cards in the deck, so there are $\binom{11}{6}$ ways this can happen (and the remaining $5$ cards must be in your opponent's hand. There are $\binom{12}{7}$ total ways the cards could be distributed between the deck and the opponent's hand if we don't care where the Ace of Diamonds is. So the probability the Ace of Diamonds is in the deck is: $$\frac{\binom{11}{6}}{\binom{12}{7}}=\frac{\frac{11!}{6!5!}}{\frac{12!}{7!5!}}=\frac{11!7!}{12!6!}= \frac{7}{12}.$$

As others have stated, your formula for Question 2 is correct, but you made an arithmetic error.

For Question 3, let $B=\{\text{other player's hand has at most one spade}\}$ and $A=\{\text{Ace of Diamonds is in the deck}\}$. As you said, you need to compute $P(B\mid A)$. Since the event $B$ is somewhat complicated - the other player's hand could have zero or one spade - but $B^c$ is simple, because it means the other player's hand has both spades. Now you can use the fact that $P(B\mid A) = 1-P(B^c \mid A)$. Now you just need to compute $P(B^c \mid A)=P(B^c \cap A)/P(A)$. The numerator involves counting the number of ways that both spades are in the opponent's hand and the Ace of Diamonds is in the deck, which I believe you should be able to do.