Problem 5
Question
You are dealt one card from a 52-card deck. Find the probability that you are not dealt a picture card.
Step-by-Step Solution
Verified Answer
The probability that a non-picture card is dealt is \( \frac{10}{13} \).
1Step 1: Determine the Number of Desired Outcomes
In a standard 52-card deck, there are 4 Kings, 4 Queens, and 4 Jacks, making a total of 12 picture cards. Subtracting this from 52 gives the total number of non-picture cards: \(52 - 12 = 40\). So there are 40 desired outcomes, which is receiving a non-picture card.
2Step 2: Calculate the Total Number of Possible Outcomes
The total number of possible outcomes is simply the number of cards in the deck, which is 52.
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. Thus, the probability of drawing a non-picture card is given by \( \frac{40}{52}\). Simplifying this fraction by dividing both the numerator and denominator by 4 results in \( \frac{10}{13} \).
Key Concepts
52-card deckPicture cardsNon-picture cardsFraction simplification
52-card deck
A 52-card deck is commonly used in many card games and contains 52 unique cards. These cards are divided into four suits:
Understanding the structure of a standard deck is important when calculating probabilities, especially in games that involve drawing cards at random. This uniform distribution across four suits makes probability calculations more straightforward.
Knowing that there are exactly 52 cards helps ensure accuracy when determining potential outcomes.
- Hearts
- Diamonds
- Clubs
- Spades
Understanding the structure of a standard deck is important when calculating probabilities, especially in games that involve drawing cards at random. This uniform distribution across four suits makes probability calculations more straightforward.
Knowing that there are exactly 52 cards helps ensure accuracy when determining potential outcomes.
Picture cards
Picture cards, also known as face cards, consist of Jacks, Queens, and Kings. Each of the four suits in a deck has these three picture cards.
When calculating the probability of drawing a particular type of card, knowing how many picture cards there are is crucial. Since picture cards make up a smaller portion of the deck, they change the dynamics of randomness in drawing cards.
- 4 Jacks
- 4 Queens
- 4 Kings
When calculating the probability of drawing a particular type of card, knowing how many picture cards there are is crucial. Since picture cards make up a smaller portion of the deck, they change the dynamics of randomness in drawing cards.
Non-picture cards
Non-picture cards include all cards in the deck that are not Jacks, Queens, or Kings. These are numbered cards from 2 to 10 and the Aces, across all four suits.
Since there are 12 picture cards in the deck, the non-picture cards count to 40, as calculated by subtracting the 12 picture cards from the total of 52 cards.
Understanding non-picture cards is important when calculating certain probabilities, such as in the exercise of finding the probability of not drawing a picture card. Knowing there are 40 non-picture cards lets us determine the likelihood of this event accurately.
Since there are 12 picture cards in the deck, the non-picture cards count to 40, as calculated by subtracting the 12 picture cards from the total of 52 cards.
Understanding non-picture cards is important when calculating certain probabilities, such as in the exercise of finding the probability of not drawing a picture card. Knowing there are 40 non-picture cards lets us determine the likelihood of this event accurately.
Fraction simplification
Fraction simplification is the process of making a fraction as simple as possible by reducing it to its lowest terms.
In probability, simplifying fractions helps make results easier to understand and interpret.
For example, in the probability calculation of drawing a non-picture card, the initial fraction is \( \frac{40}{52} \).
By finding the greatest common divisor of both the numerator and the denominator, you can simplify it to \( \frac{10}{13} \).
Simplified fractions allow for a clearer understanding of the probability and can be more easily compared with other probabilities.
In probability, simplifying fractions helps make results easier to understand and interpret.
For example, in the probability calculation of drawing a non-picture card, the initial fraction is \( \frac{40}{52} \).
By finding the greatest common divisor of both the numerator and the denominator, you can simplify it to \( \frac{10}{13} \).
Simplified fractions allow for a clearer understanding of the probability and can be more easily compared with other probabilities.
Other exercises in this chapter
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