Question

You have a standard deck of cards. The deck has 52 total cards and contains 4 suits: hearts, clubs, diamonds, and spades. Each suit consists of cards numbered 2-10, a jack, a queen, a king, and an ace.
You select one card at random from the deck. Let A be the event that the randomly selected card is a diamond and let B be the event that the card is a king. Based on this information, answer the following questions.

Answer 1

Answer:

$P(A) = \dfrac{1}{4}\\P(B) = \dfrac{1}{13}\\P(A \cap B) = \dfrac{1}{52}\\P(A/B) = \dfrac{1}{4}\\P(A/B) = P(A)$

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:

$P(E) = \dfrac{\text{Number of favorable cases}}{\text {Total number of cases}}$

For event A, number of cases possible for a card to be diamond = 13

$P(A) = \dfrac{13}{52} \\\Rightarrow P(A) = \dfrac{1}{4}$

For event B, number of cases possible for a card to be a king = 4

$P(B) = \dfrac{4}{52} \\\Rightarrow P(B) = \dfrac{1}{13}$

For the event, $A \cap B$, the card is a king and diamond, only one case is possible.

Because there is only one card for king of diamond.

$P(A \cap B) = \dfrac{1}{52}$

Formula for P(A/B):

$P(A/B) = \dfrac{P(A \cap B)}{P(B)}$

$\Rightarrow \dfrac{\dfrac{1}{52}}{\dfrac{1}{13}}\\\Rightarrow \dfrac{1}{4}$

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.