Question

a single card is drawn at random from this deck.
f) what is the probability of it being suit b?
$\frac{13}{43}$
g) what is the probability of it having rank 11?
$\frac{1}{43}$
h) 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?
$\frac{1}{13}$
i) 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?
1
j) 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 ??? are ???
we can also see this because the answers to part (g) and part ??? are ???

Explanation:

Step1: Recall probability formula

The probability of an event $E$ is $P(E)=\frac{n(E)}{n(S)}$, where $n(E)$ is the number of elements in the event $E$ and $n(S)$ is the number of elements in the sample - space.

Step2: Analyze part (f)

Assume the deck has 43 cards. If each suit has the same number of cards and there are 4 suits in a standard - like deck structure, and we assume suit $B$ has 13 cards. So $P(\text{suit }B)=\frac{13}{43}$.

Step3: Analyze part (g)

If there is only 1 card of rank 11 in the 43 - card deck, then $P(\text{rank }11)=\frac{1}{43}$.

Step4: Analyze part (h)

Among the 13 cards of suit $B$, if there is 1 card of rank 11, then using the conditional - probability formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In the context of counting, $P(\text{rank }11|\text{suit }B)=\frac{1}{13}$.

Step5: Analyze part (i)

Among the cards of rank 11 (assume there is 1 card of rank 11 in the deck), if it is of suit $B$, then $P(\text{suit }B|\text{rank }11) = 1$.

Step6: Analyze part (j)

Two events $A$ and $B$ are independent if $P(A\cap B)=P(A)\times P(B)$. Here, let $A$ be the event that the card is suit $B$ and $B$ be the event that the card has rank 11. $P(A)=\frac{13}{43}$, $P(B)=\frac{1}{43}$, and $P(A\cap B)$ (probability of being suit $B$ and rank 11) is $\frac{1}{43}$. Since $P(A)\times P(B)=\frac{13}{43}\times\frac{1}{43}
eq\frac{1}{43}=P(A\cap B)$, the events are not independent. We can see this because the answers to part (f) and part (h) are not equal (in the context of conditional - probability relationship for independence), and the answers to part (g) and part (i) are not equal.

Answer:

f) $\frac{13}{43}$
g) $\frac{1}{43}$
h) $\frac{1}{13}$
i) $1$
j) Not independent; because the answers to part (f) and part (h) are not equal, and the answers to part (g) and part (i) are not equal.