Question

a) what is the probability of it having rank 11?
1/20
c) 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?
1/20
d) 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?
1/3
e) the events, the card is suit b, 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???
a deck consists of 43 cards with 3 suits labelled a, b, and c. unlike the earlier deck, this deck has 20 cards with suit a numbered 1 to 20, 13 cards with suit b numbered 1 to 13, and it only has 10 cards with suit c numbered 1 to 10. the number of cards in each suit is not the same.
a single card is drawn at random from this deck.
f) what is the probability of it being suit b?
13/43

Explanation:

Step1: Recall probability formula

The probability of an event $E$ is given by $P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.

Step2: Solve part - c

Among the cards with suit $B$, there are 13 cards. The number of cards with rank 11 in suit $B$ is 1. So the probability of having rank 11 given that it is suit $B$ is $P(\text{rank 11}|\text{suit }B)=\frac{1}{13}$.

Step3: Solve part - d

The number of cards with rank 11 in the deck: There is 1 card with rank 11 in suit $A$, 1 in suit $B$ and 1 in suit $C$ (assuming a standard ranking concept for each suit), so there are 3 cards with rank 11 in total. The number of cards with rank 11 that are suit $B$ is 1. So $P(\text{suit }B|\text{rank }11)=\frac{1}{3}$.

Step4: Solve part - e

Two events $A$ and $B$ are independent if $P(A|B) = P(A)$ and $P(B|A)=P(B)$. Let event $A$ be "the card is suit $B$" and event $B$ be "the card has rank 11". $P(A)=\frac{13}{43}$, $P(B)=\frac{3}{43}$, $P(A|B)=\frac{1}{3}$, $P(B|A)=\frac{1}{13}$. Since $P(A|B)
eq P(A)$ and $P(B|A)
eq P(B)$, the events are not independent.

Step5: Solve part - f

The total number of cards in the deck is $n = 43$. The number of cards with suit $B$ is 13. So the probability of drawing a card of suit $B$ is $P(\text{suit }B)=\frac{13}{43}$.

Answer:

c) $\frac{1}{13}$
d) $\frac{1}{3}$
e) Not Independent
f) $\frac{13}{43}$