Question

a deck consists of 60 cards with 3 suits labelled a, b, and c, and numbered ranks from 1 to 20. that is, there are 20 cards of each suit and 3 cards of each rank.
a single card is drawn at random from this deck.
a) what is the probability of it being suit b?
1/3
b) 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?
e) the events, the card is suit b, and the card has rank 11, are ???
we can see this because the answers to part (a) and part ??? are ???

Explanation:

Step1: Calculate total number of cards

There are 60 cards in total.

Step2: Calculate probability for part (a)

There are 20 cards of suit B out of 60. So $P(\text{suit }B)=\frac{20}{60}=\frac{1}{3}$.

Step3: Calculate probability for part (b)

There are 3 cards of rank 11 out of 60. So $P(\text{rank }11)=\frac{3}{60}=\frac{1}{20}$.

Step4: Calculate probability for part (c)

Among 20 cards of suit B, there is 1 card of rank 11. So $P(\text{rank }11|\text{suit }B)=\frac{1}{20}$.

Step5: Calculate probability for part (d)

Among 3 cards of rank 11, there is 1 card of suit B. So $P(\text{suit }B|\text{rank }11)=\frac{1}{3}$.

Step6: Determine independence for part (e)

Two events A and B are independent if $P(A\cap B)=P(A)\times P(B)$. Here $P(\text{suit }B\cap\text{rank }11)=\frac{1}{60}$, $P(\text{suit }B)\times P(\text{rank }11)=\frac{1}{3}\times\frac{1}{20}=\frac{1}{60}$. So the events "the card is suit B" and "the card has rank 11" are independent.

Answer:

a) $\frac{1}{3}$
b) $\frac{1}{20}$
c) $\frac{1}{20}$
d) $\frac{1}{3}$
e) independent