Question

You shuffle a deck of 52 cards well and then expose the top 3 cards. Give exact answers expressed as simplified fractions.

(a) What is the probability that the 3 cards are from 3 different suits? Use the sample space consisting of ordered triples (x1, x2, x3), where xi is the i-th card exposed.

(b) What is the probability that the 3 cards are from 3 different suits? Use the sample space consisting of sets {x1, x2, x3} of 3 distinct cards.

(c) What is the probability that all 3 cards come from the same suit? Use the sample space consisting of ordered triples (x1, x2, x3), where xi is the i-th card exposed.

(d) What is the probability that all 3 cards come from the same suit? Use the sample space consisting of sets {x1, x2, x3} of 3 distinct cards.

Provide a brief 1-2 sentence explanation for each question.

Answer 1

(a)

Probability to select a card of any suit (let it be suit 1) = 1

Probability to select a card from the different suit as the first one (let this suit be suit 2)

= (51 - 12)/51 = 39/51 (12 cards from the suit 1 remain now jn 51 cards)

Probability to select a card from the different suit as the first and second one

= (50 - 12 - 12)/50 = 26/50 (12 cards from the suit 1 and suit 2 remain now in 50 cards)

Probability that the 3 cards are from 3 different suits = 1 * (39/51) * (26/50) = 0.3976471

(b)

Number of ways to select 3 cards from deck of 52 cards = ⁵²C₃

Number of ways to select each of 3 cards from 3 different suits (13 cards) = ¹³C₁ * ¹³C₁ * ¹³C₁

Number of ways to select 3 suits from 4 suits = ⁴C₃

Probability that the 3 cards are from 3 different suits = (Number of ways to select 3 suits from 4 suits * Number of ways to select each of 3 cards from 3 different suits ) / Number of ways to select 3 cards from deck of 52 cards

= (⁴C₃  * ¹³C₁ * ¹³C₁ * ¹³C₁ ) /  ⁵²C₃

= (4 * 13 * 13 * 13) / 22100

= 0.3976471

(c)

Probability to select a card of any suit (let it be suit 1) = 1

Probability to select a card from the same suit as the first one

= 12/51   (12 cards from the suit 1 remain now in 50 cards)

Probability to select a card from the same suit as the first and second one

= 11/50 (11 cards from the suit 1 remain now in 50 cards)

Probability that the 3 cards are from 3 different suits = 1 * (12/51) * (11/50) = 0.05176471

(d)

Number of ways to select each of cards from a given suit  = ¹³C₃

Number of ways to select a suit from 4 suits = ⁴C₁

Probability that the 3 cards are from 3 different suits = (Number of ways to select 3 suits from 4 suits * Number of ways to select each of 3 cards from 3 different suits ) / Number of ways to select 3 cards from deck of 52 cards

= (⁴C₁ * ¹³C₃ ) /  ⁵²C₃

= (4 * 286) / 22100

= 0.05176471