Question

task

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? Are these events (“ace” and “red”)
mutually exclusive? Are they independent?
e. List two events that are mutually exclusive
for a single draw from a deck of cards.
f. What is the probability of drawing a red
king? What is the probability of drawing a
face card in hearts? Are these two events
mutually exclusive? Are they independent?

Answer 1

a. What is the probability of drawing a king of any suit?

there are four kings

so required probability = 4/52 = 1/13

b. What is the probability of drawing a face card that is also a spade?

there are 3 face spade cards

so required probability is 3/52

c. What is the probability of drawing a card without a number on it?

there are 4*4 = 16 cards (each suit have 1 ace and 3 face cards)

So required probability = 16/52 = 4/13

d. What is the probability of drawing a red card? What is the probability of drawing an ace? What is the probability of drawing
red ace? Are these events (“ace” and “red”) mutually exclusive? Are they independent?

1) there are 26 red cards

so probability of drawing a red card= 26/52 = 1/2

2) There are 4 aces

So the probability of drawing an ace = 4/52 = 1/13

3) there two red aces so, the probability of drawing red ace = 2/52 = 1/26

4) No, they are not mutually exclusive

Because there intersection is not an empty set

because the two aces are also red.

5) P(Aces) = 4/52 = 1/13

P(red cards) = 26/52 = 1/2

P(Aces)*P(red cards) = 1/26

P( red cards and aces) = 2/52 = 1/26

Therefore, “ace” and “red” are independent events.

e) Drawing face card and drawing ace are mutually exclusive event.

f. What is the probability of drawing a red king? What is the probability of drawing a face card in hearts? Are these two events
mutually exclusive? Are they independent?

1) there are two red kings

so probability of drawing a red king = 2/52 = 1/26

2) there are 3 hearts of face cards

so probability of drawing a face card in hearts = 3/52

3) No because red king and king in heart is a common of these two events.

4) P( red king)*P(face card in hearts) = (1/26)*(3/52)

P( red king and face card in heart) = 1/52

Therefore they are not independent events.