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Chapter 5: Problem 5

Remove one card from an ordinary deck of 52 cards. What is the probability that it is an ace, a diamond, or black?

Short Answer

Expert verified
The probability is \( \frac{10}{13} \).

Step by step solution

01

Count Total Possible Outcomes

An ordinary deck of 52 cards contains a total of 52 possible outcomes, as there is one card being removed from the deck.
02

Determine Favorable Outcomes

To find the probability, count the favorable outcomes. There are 4 aces, 13 diamonds, and 26 black cards in a full deck. However, we must avoid double-counting specific cards. The ace of diamonds is both an ace and a diamond, and the two black aces (ace of spades and ace of clubs) are both black and aces.
03

Calculate Adjusted Favorable Outcomes

The individual counts are 4 aces + 13 diamonds + 26 black cards = 43 initial total outcomes. We then subtract the overlap: 1 ace of diamonds and 2 black aces. This gives us 43 - 3 = 40 unique favorable outcomes.
04

Compute Probability

The probability is the number of favorable outcomes divided by the total number of outcomes: \[ P = \frac{40}{52} \] This simplifies to \[ P = \frac{10}{13} \] after dividing both the numerator and the denominator by their greatest common divisor, which is 4.

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Key Concepts

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

Card Probability Problems
Card probability problems are common exercises in discrete mathematics. They involve calculating the likelihood of drawing specific cards from a standard deck. Probability is a measure of how likely an event is to occur. In this context, we can define probability as the number of favorable outcomes divided by the total possible outcomes in a deck. For instance, when asking the probability of drawing an ace, diamond, or black card, we consider how many of those cards exist in the deck compared to the total number. These types of problems help students understand basic concepts of probability, including favorable vs. possible outcomes.
Deck of Cards Mathematics
Understanding a standard deck of cards is crucial for solving probability problems in card games. A standard deck comprises 52 cards, divided equally into four suits: hearts, diamonds, clubs, and spades. Hearts and diamonds are red, while clubs and spades are black. Each suit contains 13 cards, ranging from Ace (1) to King (13). Knowing these details helps in calculating probabilities, such as determining how many cards meet certain criteria. For example, calculating the odds of drawing a specific type of card, like an ace or a diamond, involves counting those respective cards in the deck.
Combinatorial Probability
Combinatorial probability involves counting techniques to determine the likelihood of certain events, avoiding overcounting. In card problems, this comes into play when cards can fit multiple criteria, such as an ace of diamonds being both an ace and a diamond. It requires careful counting of overlaps to ensure accuracy. Techniques used include addition and subtraction principles. Here, we added the number of aces, diamonds, and black cards, then subtracted the count of overlapping cards, that is, black aces and the ace of diamonds. This adjustment gives a precise count of events satisfying any of the conditions.
Set Theory in Probability
Set theory provides a framework to solve probability problems by treating collections of objects as sets. Each type of card (aces, diamonds, black cards) can be viewed as a set. Problems often involve finding the union of these sets, meaning cards that satisfy either of the criteria. Careful attention is needed for overlaps, where a card might belong to more than one set. The principles of set intersection and union come into play, such as subtracting the repeating elements. This method ensures all possible combinations are accounted for, resulting in a clear and accurate probability outcome.

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Most popular questions from this chapter

If you roll eight dice, what is the probability that each of the numbers 1 through 6 appear on top at least once? What about with nine dice?

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