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?
b) what is the probability of it having rank 11?
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?
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 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, it only has 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.

Explanation:

Step1: Calculate probability of suit B in part a

There are 3 suits and each is equally - likely in the first deck. The total number of suits is 3. The probability of drawing a card of suit B is the number of favorable outcomes (suit B) divided by the total number of outcomes (3 suits). So $P(\text{suit B})=\frac{1}{3}$.

Step2: Calculate probability of rank 11 in part b

There are 20 ranks from 1 to 20. The probability of drawing a card with rank 11 is the number of favorable outcomes (rank 11) divided by the total number of outcomes (20 ranks). So $P(\text{rank 11})=\frac{1}{20}$.

Step3: Calculate probability of rank 11 given suit B in part c

Among the cards of suit B, there are 20 cards (ranks 1 - 20). The number of cards with rank 11 in suit B is 1. So $P(\text{rank 11}|\text{suit B})=\frac{1}{20}$.

Step4: Calculate probability of suit B given rank 11 in part d

Among the cards with rank 11, there are 3 cards (one for each suit). The number of cards of suit B with rank 11 is 1. So $P(\text{suit B}|\text{rank 11})=\frac{1}{3}$.

Step5: Check independence in part e

Two events A and B are independent if $P(A|B)=P(A)$ and $P(B|A)=P(B)$.
For the events "card is suit B" and "card has rank 11":
We know $P(\text{suit B})=\frac{1}{3}$, $P(\text{rank 11})=\frac{1}{20}$, $P(\text{rank 11}|\text{suit B})=\frac{1}{20}$, $P(\text{suit B}|\text{rank 11})=\frac{1}{3}$.
Since $P(\text{rank 11}|\text{suit B}) = P(\text{rank 11})$ and $P(\text{suit B}|\text{rank 11})=P(\text{suit B})$, the two events are independent.

Answer:

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