Question

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? a) 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. a single card is drawn at random from this deck. f) what is the probability of it being suit b? g) what is the probability of it having rank 11? 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? 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? j) the events, the card is suit b, and the card has rank 11, are ???

Explanation:

Step1: Calculate total number of cards

The deck has 20 cards of suit A, 13 cards of suit B and 10 cards of suit C. So the total number of cards $N=20 + 13+10=43$.

Step2: Probability of suit B

The number of suit - B cards $n_B = 13$. The probability of drawing a suit - B card $P(B)=\frac{n_B}{N}=\frac{13}{43}$.

Step3: Probability of rank 11

Only suit B has cards numbered from 1 - 13 and there is 1 card with rank 11 in suit B. So the probability of drawing a card with rank 11 $P(11)=\frac{1}{43}$.

Step4: Probability of rank 11 given suit B

Among the 13 suit - B cards, there is 1 card with rank 11. By the formula for conditional probability $P(11|B)=\frac{n(11\cap B)}{n(B)}$, where $n(11\cap B) = 1$ and $n(B)=13$, so $P(11|B)=\frac{1}{13}$.

Step5: Probability of suit B given rank 11

Among the cards with rank 11, there is 1 card which is of suit B. So $P(B|11)=\frac{n(11\cap B)}{n(11)}=\frac{1}{1}$.

Step6: Check for independence

Two events A and B are independent if $P(A\cap B)=P(A)\times P(B)$. Here, $P(B)=\frac{13}{43}$, $P(11)=\frac{1}{43}$, and $P(B\cap 11)=\frac{1}{43}$. Since $P(B)\times P(11)=\frac{13}{43}\times\frac{1}{43}
eq\frac{1}{43}=P(B\cap 11)$, the events are not independent.

Answer:

f) $\frac{13}{43}$
g) $\frac{1}{43}$
h) $\frac{1}{13}$
i) 1
j) Not independent