Question

Suppose you are dealt 5 random cards from a standard deck of 52 cards, where all cards are equally likely to appear.

(a) What is your outcome space?

(b) What is the probability that you receive the ace of hearts?

(c) Let AH be the event that you receive the ace of hearts, AC the event that you receive the ace of clubs, AD the event that you receive the ace of diamonds, and AS the event that you receive the ace of spades. If A is the event that you receive at least one ace, write A in terms of AH, AC, AD, and AS.

(d) Use the union bound to give an upper bound on the probability of A.

Answer 1

(a)

The outcome space is the five different cards from a standard deck of 52 cards. For example, ace of hearts, 2 of spades, 3 of diamonds, queen of spades, 10 of diamonds.

(b)

Number of ways to draw 5 random cards from a standard deck of 52 cards = ⁵²C₅ = 2598960

Let we have selected one of the card as ace of hearts. Then, number of ways to draw remaining 4 random cards from a standard deck of remaining 51 cards = ⁵¹C₄ = 249900

Probability that you receive the ace of hearts = 249900 / 2598960 = 0.09615385

(c)

A is the event that you receive at least one ace. Then, A is the event present in all events AH, AC, AD and AS. Also, event A will not lie outside of the union of events AH, AC, AD and AS.

Thus,

A = AH AC AD AS

(d)

From part (b), P(AH) = 0.09615385

Similarly, we can show that P(AC) = P(AD) = P(AS) = 0.09615385

Using union bound,

P(A) = P(AH AC AD AS) P(AH) + P(AC) + P(AD) + P(AS) = 0.09615385 + 0.09615385 + 0.09615385 + 0.09615385

=> P(A) 0.3846154

The upper bound on the probability of A is 0.3846154.