Question

a) what is the probability of it having rank 11?
b) what is the probability of it having rank 11 given that it is suit b?
hint: among the cards with suit b, how many of them have rank 11?
c) what is the probability of it being suit b given that it has rank 11?
hint: among the cards with rank 11, how many of them are suit b?
d) the events, the card is suit b, and the card has rank 11, are not independent
we can see this because the answers to part (f) and part (h) are not equal
we can also see this because the answers to part (g) and part (i) are not equal

Explanation:

Step1: Analyze part g

Assume a standard - like card - deck situation. If the total number of cards is 48 (for example, if we have some non - standard deck setup), and there are 2 cards with rank 11, the probability of having rank 11 is $\frac{2}{48}=\frac{1}{24}$. But here it seems the total number of cards is 48 and the number of cards with rank 11 is 2, so $P(\text{rank }11)=\frac{2}{48}=\frac{1}{24}$.

Step2: Analyze part h

If suit B has 13 cards and only 1 of them has rank 11, then the conditional probability $P(\text{rank }11|\text{suit }B)=\frac{1}{13}$.

Step3: Analyze part i

If there are 2 cards with rank 11 and 1 of them is suit B, then the conditional probability $P(\text{suit }B|\text{rank }11)=\frac{1}{2}$.

Step4: Analyze part j

Two events A and B are independent if $P(A|B)=P(A)$ and $P(B|A)=P(B)$. Here, for event A: card is suit B and event B: card has rank 11. Since $P(\text{rank }11|\text{suit }B)
eq P(\text{rank }11)$ and $P(\text{suit }B|\text{rank }11)
eq P(\text{suit }B)$, the events are not independent.

Answer:

a) $\frac{2}{48}$ (assuming 48 - card deck and 2 cards of rank 11)
b) $\frac{1}{13}$
c) $\frac{1}{2}$
d) Not independent
e) h; not equal
f) i; not equal