Two cards are selected from a standard deck of 52 playing cards. The first card is not replaced before the second card is selected. Find the probability of selecting a six and then selecting a seven. The probability of selecting a six and then selecting a seven is (Round to three decimal places as needed.)
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: Understanding Probability with Playing Cards**
**Problem Statement:**
Two cards are selected from a standard deck of 52 playing cards. The first card is not replaced before the second card is selected. Find the probability of selecting a six and then selecting a seven.
**Question:**
- What is the probability of selecting a six and then selecting a seven?
- The probability should be rounded to three decimal places.
**Solution Explanation:**
To determine the probability of two sequential events where one event affects the other (since the first card is not replaced), use the concept of conditional probability.
### Step-by-Step Calculation:
1. **Probability of Selecting a Six (First Draw):**
- There are 4 sixes in a standard deck of 52 cards.
- Thus, the probability of drawing a six is: \( P(6) = \frac{4}{52} \).
2. **Probability of Selecting a Seven (Second Draw):**
- After drawing a six, there are now 51 cards left in the deck.
- There are still 4 sevens in the deck.
- Thus, the probability of drawing a seven after drawing a six is: \( P(7 | 6) = \frac{4}{51} \).
3. **Joint Probability:**
- The probability of both events occurring in sequence (drawing a six first and then drawing a seven) is the product of the individual probabilities.
- So, the probability is: \( P(6 \text{ then } 7) = \frac{4}{52} \times \frac{4}{51} \).
### Calculation:
\[ P(6) = \frac{4}{52} = \frac{1}{13} \]
\[ P(7 | 6) = \frac{4}{51} \]
\[ P(6 \text{ then } 7) = \frac{1}{13} \times \frac{4}{51} = \frac{4}{663} \approx 0.006 \]
### Conclusion:
The probability of selecting a six and then selecting a seven, without replacement, rounded to three decimal places is:
\[ 0.006 \]
**Note:** Understanding the role of conditional probability and how the first event affects the second is crucial in solving such problems. This example helps illustrate the concept of dependent events in probability theory.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe479fc91-2cd6-4fb6-bfec-c8ec0e0a9b03%2F664fefb1-7fb9-44cb-a85a-fd1cc8c7ba74%2Faxugyz_processed.jpeg&w=3840&q=75)
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