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Chapter 2: Problem 120

Four cards are to be dealt successively, at random and without replacement, from an ordinary deck of playing cards. Find the probability of receiving a spade, a heart, a diamond, and a club, in that order.

Short Answer

Expert verified
The probability of receiving a spade, a heart, a diamond, and a club, in that order, when drawing 4 cards without replacement from an ordinary deck of playing cards is approximately 0.00264 or 0.264%.

Step by step solution

01

Probability of drawing a spade first

There are 13 spades in the deck and 52 cards in total, so the probability of drawing a spade first is the ratio of the number of spades to the number of total cards: \[ P(S_1) = \frac{13}{52} \]
02

Probability of drawing a heart next

After drawing a spade, there are now 51 cards left in the deck, and there are still 13 hearts. So, the probability of drawing a heart at this step is the ratio of the number of hearts to the remaining number of cards: \[ P(H_2|S_1) = \frac{13}{51} \]
03

Probability of drawing a diamond after that

Once we have drawn a spade and a heart, there are now 50 cards left in the deck, and there are still 13 diamonds. The probability of drawing a diamond at this step is the ratio of the number of diamonds to the remaining number of cards: \[ P(D_3|S_1H_2) = \frac{13}{50} \]
04

Probability of drawing a club last

After drawing a spade, a heart, and a diamond, there are now 49 cards left in the deck, and there are still 13 clubs. The probability of drawing a club at this step is the ratio of the number of clubs to the remaining number of cards: \[ P(C_4|S_1H_2D_3) = \frac{13}{49} \]
05

Find the overall probability

Finally, we will multiply the probabilities found in the previous steps to find the overall probability of drawing a spade, a heart, a diamond, and a club in the given order: \[ P(S_1H_2D_3C_4) = P(S_1) \times P(H_2|S_1) \times P(D_3|S_1H_2) \times P(C_4|S_1H_2D_3) \\ = \frac{13}{52} \times \frac{13}{51} \times \frac{13}{50} \times \frac{13}{49} \] Using a calculator, we find the final probability to be: \[ P(S_1H_2D_3C_4) \approx 0.00264 \] So, there is approximately a 0.264% chance of receiving a spade, a heart, a diamond, and a club, in that order, from an ordinary deck of playing cards when drawing 4 cards without replacement.

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