A normal deck of cards has 52 cards, consisting of 13 each of four suits: spades, hearts, diamonds, and clubs. Hearts and diamonds are red, while spades and clubs are black. Each suit has an ace, nine cards numbered 2 through 10, and three "face cards." The face cards are a jack, a queen, and a king. Answer the following questions for a single card drawn at random from a well-shuffled deck of cards. a. What is the probability of drawing a king of any suit? b. What is the probability of drawing a face card that is also a spade? c. What is the probability of drawing a card without a number on it? d. What is the probability of drawing a red card? What is the probability of drawing an ace? What is the probability of drawing a red ace? Arc these events ("ace" and "red") mutually exclusive? Are they independent? List two events that are mutually exclusive

a) Probability of drawing a king of any suit can be found out by first counting the total number of kinds in the set of 52 cards. Each suit has 1 king and there are 4 suits so, there are a total of 4 kings in the deck.

Probability (King) = No. of Kings/Total number of cards

= 4/52

Probability (King) = 1/13

b) Face cards that are also a spade are only 3 out of the whole deck. Hence, the probability can be computed as:

P(face cards of spades) = No. of face cards of spades/Total no. of cards

P(face cards of spades) = 3/52

c) Card without numbers are the face cards plus aces. So, each suit has 3 face cards and 1 ace i.e. 4 cards without numbers and there are 4 different suits so, total number of cards without numbers are: 4 x 4 = 16 cards.

P(cards without numbers) = No. of cards without numbers/Total no. of cards

= 16/52

P(cards without numbers) = 4/13

d) The red cards in the deck are all the hearts and diamonds. So, the total number of red cards = 13 + 13 = 26

P(Red Card) = No. of red cards/Total no. of cards

= 26/52

P(Red Card) = 1/2

There is 1 ace in each of the suits hence we have a total of 4 aces in the deck.

P(Ace) = No. of aces/Total no. of cards

= 4/52

P(Ace) = 1/13

Number of red aces in the deck are 2, one from hearts and one from diamonds. So,

P(Red Ace) = No. of red aces/Total no. of aces

= 2/52

P(Red Ace) = 1/26

For two events A and B to be mutually exclusive, they must satisfy the equation:

P(A or B) = P(A) + P(B)

These events have nothing in common and they cannot occur at the same time. So, Ace and Red can not be mutually exclusive because they do occur at the same time. The events ace and red are not mutually exclusive.

For two events A and B to be independent, they must satisfy the equation:

P(A and B) = P(A) x P(B)

P(Red and Ace) = P(Red) x P(Ace)

1/26 = 1/2 x 1/13

1/26 = 1/26

Hence, we can conclude that the events ace and red are independent.

Two events that are mutually exclusive can be:

1) Drawing an Ace and drawing a King. These events can not occur at the same time so we can say that these events are mutually exclusive.

2) Drawing a red card and drawing a black card. There is no such card which is red and black at the same time so, these two events can not occur at the same time hence, we can say that they are mutually exclusive.