Question

c) what is the probability of it having rank 11 given that it is suit h? hint: among the cards with suit h, how many of them have rank 11?
d) what is the probability of it being suit h given that it has rank 11? hint: among the cards with rank 11, how many of them are suit h?
e) the events, the card is suit h, and the card has rank 11, are independent we can see this because the answers to part (a) and part ??? are ??? we can also see this because the answers to part (b) and part ??? are ???

Explanation:

Step1: Recall probability - given formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. For a standard deck of 52 - card problems, we can also use the ratio of favorable cases.

Step2: Analyze part b

Let's assume a standard deck of 52 cards. There are 13 cards of each suit. Among the cards of a particular suit (say suit $B$), if we want to find the probability of having rank 11 (Jack). Since there is 1 Jack in each suit, and there are 13 cards in a suit, the probability $P(\text{rank }11|\text{suit }B)=\frac{1}{13}$.

Step3: Analyze part c

Among the 4 Jacks (rank 11) in the deck, there is 1 Jack of each suit. So if we know the card has rank 11, and we want to find the probability it is a particular suit (say suit $B$), the probability $P(\text{suit }B|\text{rank }11)=\frac{1}{4}$.

Step4: Analyze independence

Two events $A$ (card is suit $B$) and $B$ (card has rank 11) are independent if $P(A|B) = P(A)$ and $P(B|A)=P(B)$.
The probability that a card is a particular suit $P(A)=\frac{13}{52}=\frac{1}{4}$. The probability that a card has rank 11 $P(B)=\frac{4}{52}=\frac{1}{13}$.
From part b, $P(\text{rank }11|\text{suit }B)=\frac{1}{13}=P(\text{rank }11)$. From part c, $P(\text{suit }B|\text{rank }11)=\frac{1}{4}=P(\text{suit }B)$.
We can see this because the answers to part (b) and part (c) are equal to the unconditional probabilities of having rank 11 and being suit $B$ respectively.

Answer:

For part b: $\frac{1}{13}$
For part c: $\frac{1}{4}$
For part e: The blanks should be filled as follows:
The answers to part (a) and part (c) are equal.
The answers to part (b) and part (d) are equal. And the events are independent.