A hand containing exactly one pair consists of two cards with the same denomination but different suits and three more cards each with a denomination different than the denomination of the pair. For example, a hand containing one pair could be (4H, 4S, 3C,5H,King H). The 4 of hearts and the 4 of spades is the pair in the this example. Compute the probability of a hand containing exactly one pair (e.g. not two pairs, no three of a kind, etc.).
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![**Title: Probability of Drawing Exactly One Pair from a Standard Deck of Cards**
**Introduction:**
In a standard deck of 52 playing cards, each card is uniquely identified by one of four suits (hearts, clubs, diamonds, spades) and one of 13 face values or denominations (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king). A hand consists of 5 cards chosen from these 52.
**Defining the Hand with Exactly One Pair:**
A hand containing exactly one pair consists of:
- Two cards of the same denomination but different suits.
- Three other cards, each with a denomination different from the pair and each other.
**Example:**
A hand containing one pair could be:
- 4H (4 of hearts)
- 4S (4 of spades)
- 3C (3 of clubs)
- 5H (5 of hearts)
- KH (king of hearts)
In this example, the pair is formed by the 4 of hearts and the 4 of spades.
**Objective:**
Compute the probability of drawing a hand that contains exactly one pair (not two pairs, no three of a kind, etc.).
**Calculation Details:**
1. **Denominator (Total Ways to Choose 5 Cards from 52):**
The total number of ways to choose 5 cards from 52 is given by the combination formula \( \binom{52}{5} \).
\[
\binom{52}{5} = \frac{52!}{5!(52-5)!} = 2,598,960
\]
2. **Numerator (Ways to Form Exactly One Pair):**
To find the numerator, we proceed using the multiplication rule:
- **Step 1: Choose the denomination for the pair.**
There are 13 possible denominations, so 13 ways.
- **Step 2: Choose 2 out of 4 suits for this denomination.**
For each denomination chosen, there are \( \binom{4}{2} = 6 \) ways.
- **Step 3: Choose 3 different denominations for the remaining cards.**
After choosing the denomination of the pair, 12 denominations remain. We choose 3 out of these 12,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00a48d79-f805-418c-a679-1c91878d1d75%2F9b0928ef-6da6-44a7-b6e3-053c3bced4d4%2Ft5yy91_processed.png&w=3840&q=75)
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