- The probability of drawing a face card is \(\frac{12}{52} = \boxed{\frac{3}{13}}\).
- The probability of drawing a red 10 is \(\frac{2}{52} = \boxed{\frac{1}{26}}\).
- The probability of drawing a card that is not a diamond is \(\frac{39}{52} = \boxed{\frac{3}{4}}\).
Explanation
- Analyze the composition of a standard 52-card deck
A standard deck of cards contains a total of \(52\) cards. These are divided into four suits: hearts, diamonds, clubs, and spades, each with \(13\) ranks ranging from Ace (A) to 10, and then the face cards: Jack (J), Queen (Q), and King (K). Hearts and diamonds are red, while clubs and spades are black. We will calculate the probability of specific events by finding the ratio of the number of favorable outcomes to the total number of possible outcomes, which is \(52\).
- Calculate the probability of drawing a face card
Face cards consist of the Jack, Queen, and King for each of the four suits. Since there are \(3\) face cards in each suit and \(4\) suits in total, the number of face cards is:
\[3 \times 4 = 12\]
The probability \(P(\text{face card})\) is the ratio of the number of face cards to the total number of cards:
\[P(\text{face card}) = \frac{12}{52}\]
We simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is \(4\):
\[\frac{12 \div 4}{52 \div 4} = \frac{3}{13}\]
- Calculate the probability of drawing a red 10
There are two red suits: hearts and diamonds. In each of these suits, there is exactly one '10'. Therefore, there is one 10 of hearts and one 10 of diamonds, making a total of two red 10s. The probability \(P(\text{a red 10})\) is:
\[P(\text{a red 10}) = \frac{2}{52}\]
Simplifying this fraction by dividing by \(2\):
\[\frac{2 \div 2}{52 \div 2} = \frac{1}{26}\]
- Calculate the probability of drawing a card that is not a diamond
There are \(13\) diamonds in a standard deck. To find the number of cards that are "NOT a diamond," we subtract the number of diamonds from the total number of cards:
\[52 - 13 = 39\]
The probability \(P(\text{NOT a diamond})\) is:
\[P(\text{NOT a diamond}) = \frac{39}{52}\]
We simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is \(13\):
\[\frac{39 \div 13}{52 \div 13} = \frac{3}{4}\]
Examples
This type of probability calculation is essential in games of chance, such as poker or bridge, where players need to assess the likelihood of receiving specific cards. For example, if you are playing a game where you need a heart or a spade (not a diamond), you can quickly calculate your odds using these methods to make better strategic decisions during your turn. Understanding these fundamental probabilities helps in any scenario involving random sampling from a fixed set.