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Chapter 4: Problem 11

If you pick a card from a standard deck of 52 cards, what is the probability of getting a. a \(7 ?\) b. a club? c. a spade or a club? d. a diamond or a \(5 ?\)

Short Answer

Expert verified
a. \( \frac{1}{13} \), b. \( \frac{1}{4} \), c. \( \frac{1}{2} \), d. \( \frac{4}{13} \).

Step by step solution

01

Total Possible Outcomes

A standard deck of 52 cards has 52 possible outcomes when picking one card at random.
02

Probability of Getting a 7

There are 4 sevens in a standard deck (one for each suit: hearts, diamonds, clubs, and spades). Therefore, the probability of drawing a 7 is calculated using the formula \( P(7) = \frac{\text{Number of 7s}}{\text{Total cards}} = \frac{4}{52} = \frac{1}{13} \).
03

Probability of Getting a Club

There are 13 clubs in a deck (one per rank from Ace to King). Thus, the probability of drawing a club is \( P(\text{club}) = \frac{13}{52} = \frac{1}{4} \).
04

Probability of Getting a Spade or a Club

Since spades and clubs are two separate suits, and there are 13 of each, we add the probabilities: \( P(\text{spade or club}) = \frac{13}{52} + \frac{13}{52} = \frac{26}{52} = \frac{1}{2} \).
05

Probability of Getting a Diamond or a 5

There are 13 diamonds, and 4 fives, but we have counted the 5 of diamonds twice. So, \( P(\text{diamond or 5}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13} \).

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Key Concepts

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

Standard Deck of Cards
A standard deck of cards is common in many classic games and consists of 52 unique cards. These cards are split into four suits: Hearts, Diamonds, Clubs, and Spades. Each suit contains 13 cards, which range from Ace through King. This division results in every suit having:
  • 1 Ace
  • Numbered cards from 2 to 10
  • 1 Jack, 1 Queen, and 1 King
Understanding this structure is crucial for probability calculations involving card games. When you pick a card at random from a complete deck, you have a total of 52 potential outcomes. This foundational setup allows players and mathematicians alike to calculate various probabilities, such as drawing certain ranks or suits from the deck.
Combinatorics
Combinatorics is a branch of mathematics focused on counting, arranging, and finding patterns. In the context of a deck of cards, combinatorics is used to determine the number of ways a particular card or combination of cards can be drawn. For example, in the exercise provided, you might be interested in learning about the number of sevens or the number of clubs in the deck. Combinatorics helps in counting these possibilities:
  • There are 4 sevens (one in each suit).
  • Each suit (e.g., clubs) contains 13 cards.
Using these counts, you can then calculate probabilities using basic combinatorial formulas. Like for the probability of drawing either a spade or a club, we use the total number of spades and clubs divided by the total number of cards in the deck.
Mathematical Calculations
Mathematical calculations in probability involve determining the likelihood of an event occurring. For card probability, this often means dividing the number of favorable outcomes by the total number of possible outcomes.Let's look at some typical calculations from the exercise:
  • Probability of getting a 7: With 4 sevens in the deck, the probability formula is \( P(7) = \frac{4}{52} = \frac{1}{13} \).
  • Probability of getting a club: As there are 13 clubs, the probability is \( P(\text{club}) = \frac{13}{52} = \frac{1}{4} \).
  • Probability of a spade or a club: Calculated by adding the probabilities of these separate suits: \( P(\text{spade or club}) = \frac{13}{52} + \frac{13}{52} = \frac{26}{52} = \frac{1}{2} \).
These calculations show how understanding card deck structure and simple arithmetic can lead to knowing the chance of drawing certain cards.

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Most popular questions from this chapter

The following contingency table provides data from a sample of 6,224 individuals who were exposed to smallpox in Boston. $$ \begin{array}{|c|c|c|c|} \hline & \text { Inoculated } & \text { Not Inoculated } & \text { Total } \\ \hline \text { Lived } & 238 & 5136 & 5374 \\ \hline \text { Died } & 6 & 844 & 850 \\ \hline \text { Total } & 244 & 5980 & 6224 \\ \hline \end{array} $$ a. What is the probability that a person was inoculated? b. What is the probability that a person lived? c. What is the probability that a person died or was inoculated? d. What is the probability that a person died given they were inoculated? e. What is the probability that a person died given they were not inoculated? f. Does it appear that survival depended on if a person were inoculated? Or are they independent? Use probability to support your claim.

A bag contains 3 green marbles, 4 red marbles, and 5 blue marbles (and no others). If you randomly pull out a marble and put the marble back 3 times, what is the probability that you pull out a blue marble all 3 times? Write the probability in all three forms.

There is a \(15 \%\) chance that a shopper entering a computer store will purchase a computer, a \(25 \%\) chance they will purchase a game/software, and there is a \(10 \%\) chance they will purchase both a computer and a game/software. a. Create a contingency table for the information. b. What is the probability that a shopper will not purchase a computer and will not purchase a game/software? c. What is the probability that a shopper will purchase a computer or purchase a game/software? d. What is the probability that a shopper will purchase a game/software given they have purchased a computer? e. What is the probability that a shopper will purchase a game/software given they did not purchase a computer? f. Does it appear that purchasing a game/software depends on whether the shopper purchased a computer? Or are they independent? Use probability to support your claim.

A box contains four black pieces of cloth, two striped pieces, and six dotted pieces. A piece is selected randomly and then placed back in the box. A second piece is selected randomly. What is the probability that: a. both pieces are dotted? b. the first piece is black, and the second piece is dotted? c. one piece is black, and one piece is striped?

After finishing the course, among the 32 students in a Math 105 class, 25 could successfully construct a contingency table, 27 passed the class, and 29 could successfully construct a contingency table or passed the class. a. Create a contingency table for the information. b. What is the probability that a student passed and could not successfully eonstruct a contingency table? c. What is the probability that a student passed given they could not successfully construct a contingency table?
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