Problem 17
Question
You are dealt one card from a standard 52 card deck. Find the probability of being dealt: a picture card.
Step-by-Step Solution
Verified Answer
The probability of being dealt a picture card from a standard deck is approximately 0.231, or 23.1%.
1Step 1: Identify the Total Number of Outcomes
A standard deck of 52 cards is comprised of 4 suits (Hearts, Diamonds, Clubs, Spades), each with 13 ranks (Ace through 10, and the three picture cards Jack, Queen, King). Thus, there are a total of 52 possible outcomes when drawing one card.
2Step 2: Identify the Desired Outcomes
The picture cards are the King, Queen, and Jack. Each of the 4 suits includes one each of these, so there are 3 picture cards per suit, and thus 12 (3 per suit * 4 suits) picture cards in total.
3Step 3: Calculate the Probability
The probability of an event is calculated as the number of desired outcomes divided by the total number of outcomes. In this case, the probability (P) of drawing a picture card is calculated as P = number of picture cards / total number of cards = 12 / 52 = 0.230769, or approximately 0.231 when rounded to three decimal places.
Key Concepts
Standard Deck of CardsPicture Card ProbabilityCalculating Probability
Standard Deck of Cards
Understanding the composition of a standard deck of cards is essential when calculating probabilities in card games or exercises. A standard deck contains 52 cards, divided into four suits: Hearts, Diamonds, Clubs, and Spades. Each suit is evenly represented in the deck and is composed of 13 ranks. These are the numbered cards from 2 to 10, the Ace, and three picture cards – the Jack, Queen, and King.
When dealing with probability and a standard deck of cards, it's important to remember that each suit is identical in composition but different in the suit symbol. For instance, a King of Hearts is a different card from a King of Spades, but for many probability problems, what matters is the rank (King, in this case) rather than the suit.
When dealing with probability and a standard deck of cards, it's important to remember that each suit is identical in composition but different in the suit symbol. For instance, a King of Hearts is a different card from a King of Spades, but for many probability problems, what matters is the rank (King, in this case) rather than the suit.
Picture Card Probability
When we talk about picture card probability, we are focused on the likelihood of drawing a Jack, Queen, or King from a standard deck. There are 4 of each of these picture cards in the deck, one in each suit – totaling 12 picture cards out of 52.
To provide an idea of perspective, imagine laying out all the cards and picking one at random. The odds of picking a picture card are like finding one particular dozen out of a total of 52 possible choices.
To provide an idea of perspective, imagine laying out all the cards and picking one at random. The odds of picking a picture card are like finding one particular dozen out of a total of 52 possible choices.
Distinguishing Picture Cards
It is important to note that many beginners tend to confuse picture cards with 'face cards,' a term sometimes used to refer to the same, but can also include the Aces depending on interpretation. For the sake of clarity in probability calculation, it always helps to explicitly state that picture cards refer strictly to Jacks, Queens, and Kings.Calculating Probability
Probability, in mathematical terms, is about measuring the chance that a particular event will occur. This is usually expressed as a fraction or a decimal.
The basic formula to remember for any simple probability is:
\[ P(E) = \frac{\text{Number of ways event E can occur}}{\text{Total number of possible outcomes}} \]
In the case of our picture card probability, the event E is drawing a picture card. There are 12 ways this can occur (since we have 12 picture cards), and there are 52 possible outcomes (one for each card in the deck). Therefore, the probability of drawing a picture card is 12/52. This fraction can also be reduced to 3/13 or converted into a decimal, approximately 0.231. Remember, expressing the probability in its simplest form often makes it easier to understand at a glance.
The basic formula to remember for any simple probability is:
\[ P(E) = \frac{\text{Number of ways event E can occur}}{\text{Total number of possible outcomes}} \]
In the case of our picture card probability, the event E is drawing a picture card. There are 12 ways this can occur (since we have 12 picture cards), and there are 52 possible outcomes (one for each card in the deck). Therefore, the probability of drawing a picture card is 12/52. This fraction can also be reduced to 3/13 or converted into a decimal, approximately 0.231. Remember, expressing the probability in its simplest form often makes it easier to understand at a glance.
Other exercises in this chapter
Problem 16
Write the first four terms of each sequence. $$a_{1}=2 \text { and } a_{n}=5 a_{n-1} \text { for } n \geq 2$$
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Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(x^{2}+2 y\right)^{4} $$
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In Exercises \(11-30,\) use mathematical induction to prove that each statement is true for every positive integer \(n\) $$ 1+2+2^{2}+\cdots+2^{n-1}=2^{n}-1 $$
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In Exercises \(17-20,\) does the problem involve permutations or combinations? Explain your answer. (It is not necessary to solve the problem. \()\) A medical r
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