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Chapter 4: Problem 13

You draw two cards from a standard deck of 52 cards without replacing the first one before drawing the second. (a) Are the outcomes on the two cards independent? Why? (b) Find \(P(\) ace on 1 st card and king on 2 nd). (c) Find \(P(\) king on 1 st card and ace on 2 nd ). (d) Find the probability of drawing an ace and a king in either order.

Short Answer

Expert verified
(a) No, dependent events. (b) \(\frac{4}{663}\). (c) \(\frac{4}{663}\). (d) \(\frac{8}{663}\).

Step by step solution

01

Understand independence of card outcomes

To decide if drawing two cards without replacement are independent events, we determine how the first draw affects the second draw. In this case, the result of the first draw influences the probability of the second draw (since the deck now has only 51 cards left). Thus, the outcomes are not independent, as the probability of drawing a king (or any card) changes based on the first card drawn.
02

Calculate probability for ace first, king second

To find the probability of drawing an ace first, then a king, start by calculating the probability of drawing an ace. There are 4 aces in a 52-card deck, so \(P(\text{ace on 1st}) = \frac{4}{52}\). After drawing an ace, the deck is reduced to 51 cards with 4 kings remaining. Thus, \(P(\text{king on 2nd | ace on 1st}) = \frac{4}{51}\). Multiply these probabilities: \[P(\text{ace on 1st and king on 2nd}) = \frac{4}{52} \times \frac{4}{51} = \frac{16}{2652} = \frac{4}{663}.\]
03

Calculate probability for king first, ace second

Similarly, calculate the probability of drawing a king first, then an ace. The initial probability of drawing a king is \(P(\text{king on 1st}) = \frac{4}{52}\). With a king out, there are still 4 aces but only 51 cards left, so \(P(\text{ace on 2nd | king on 1st}) = \frac{4}{51}\). Multiply these probabilities: \[P(\text{king on 1st and ace on 2nd}) = \frac{4}{52} \times \frac{4}{51} = \frac{16}{2652} = \frac{4}{663}.\]
04

Probability of ace and king in either order

To find the total probability of drawing an ace and a king in either order, sum up the probabilities from Step 2 and Step 3. Since these are mutually exclusive events, add the probabilities: \[P(\text{ace and king in either order}) = \frac{4}{663} + \frac{4}{663} = \frac{8}{663}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dependent Events
When dealing with probability, events are classified as dependent or independent. Dependent events are those where the outcome or occurrence of the first event affects the outcome or occurrence of the second event. In the given exercise, drawing two cards from a deck without replacement is a classic example of dependent events.
Let's say you draw a card from a deck and it's an ace. You've now changed the composition of the deck by removing one ace, leaving only 51 cards. This change affects the probability of drawing any specific card as the second card.
  • The probability of the second card being an ace or any other specific card has altered because the deck is not full after the first draw.
  • In simpler terms, since the first event influences the second, these events are not independent.
Card Probability
To understand card probability, we need to calculate specific outcomes involving the sequence of cards drawn from a deck. Imagine a standard deck of 52 cards, with each card having an equal chance of being drawn initially.

For example, considering drawing an ace on the first draw, you have four aces in 52 cards, so the probability for this is \(P(\text{ace on 1st}) = \frac{4}{52}\). Once an ace is drawn and not replaced, the probability of drawing a king next becomes \(P(\text{king on 2nd | ace on 1st}) = \frac{4}{51}\). This showcases how already drawn cards affect subsequent probabilities.
  • Always remember that each draw affects the number of remaining cards and consequently the probability of future draws.
  • The overall probability of multiple sequential events is found by multiplying their individual probabilities.
Mutually Exclusive Events
Mutually exclusive events are those that cannot happen at the same time. In other words, the occurrence of one event prevents the occurrence of another.

When calculating the probability of drawing an ace and a king in either order from a deck, each sequence (ace first, king second and king first, ace second) is considered a mutually exclusive event. Each event's occurrence rules out the other happening simultaneously.
  • This principle allows us to add probabilities of independent sequences to find a combined probability.
  • In this exercise, we added the probabilities of both scenarios: \([\frac{4}{663} + \frac{4}{663} = \frac{8}{663}]\) to find the total probability of drawing an ace and a king in any order.

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Most popular questions from this chapter

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