Answer:
[tex]P(A) = \dfrac{1}{4}\\P(B) = \dfrac{1}{13}\\P(A \cap B) = \dfrac{1}{52}\\P(A/B) = \dfrac{1}{4}\\P(A/B) = P(A)\\[/tex]
A and B are not independent events.
Step-by-step explanation:
Total number of possibilities is 52 (Total number of cards in the deck).
Formula for probability of an event E can be observed as:
[tex]P(E) = \dfrac{\text{Number of favorable cases}}{\text {Total number of cases}}[/tex]
For event A, number of cases possible for a card to be diamond = 13
[tex]P(A) = \dfrac{13}{52} \\\Rightarrow P(A) = \dfrac{1}{4}[/tex]
For event B, number of cases possible for a card to be a king = 4
[tex]P(B) = \dfrac{4}{52} \\\Rightarrow P(B) = \dfrac{1}{13}[/tex]
For the event, [tex]A \cap B[/tex], the card is a king and diamond, only one case is possible.
Because there is only one card for king of diamond.
[tex]P(A \cap B) = \dfrac{1}{52}[/tex]
Formula for P(A/B):
[tex]P(A/B) = \dfrac{P(A \cap B)}{P(B)}[/tex]
[tex]\Rightarrow \dfrac{\dfrac{1}{52}}{\dfrac{1}{13}}\\\Rightarrow \dfrac{1}{4}[/tex]
Yes, P(A) is same as P(A/B).
Here, A and B are not independent events because they have a common case i.e. a king is there which is of diamond in the deck.
