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

1/4

1/13

1/52

Yes, events a and b are independent events

Answer 2

`P(A) = (1)/(4)\\P(B) = (1)/(13)\\P(A \cap B) = (1)/(52)\\P(A/B) = (1)/(4)\\P(A/B) = P(A)\\`

A and B are not independent events.

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) = \frac{\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) = (13)/(52) \\\Rightarrow P(A) = (1)/(4)`

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

`P(B) = (4)/(52) \\\Rightarrow P(B) = (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) = (1)/(52)`

Formula for P(A/B):

`P(A/B) = (P(A \cap B))/(P(B))`

`\Rightarrow ((1)/(52))/((1)/(13))\\\Rightarrow (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.