5. Clarice has 7 new CDs; 3 are classical music and the rest are pop music. If she randomly grabs 3 CDs to listen to in the car on her way to school, what is the probability that she will select only classical music?
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.
![**Probability and Combinatorics in Real-life Scenarios**
### Problem Statement
Clarice has 7 new CDs; 3 are classical music and the rest are pop music. If she randomly grabs 3 CDs to listen to in the car on her way to school, what is the probability that she will select only classical music?
### Detailed Breakdown
To find the probability that Clarice will select only classical music CDs, we can follow these steps:
1. **Identify the Total Number of CDs**:
- Total CDs: 7
- Classical CDs: 3
- Pop CDs: 4 (since \(7 - 3 = 4\))
2. **Determine the Total Possible Combinations of 3 CDs from 7**:
The total number of ways to choose 3 CDs out of 7 is given by the combination formula \( \binom{n}{k} \):
\[
\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]
3. **Determine the Number of Favorable Combinations**:
The number of ways to choose 3 classical CDs out of 3 is:
\[
\binom{3}{3} = 1
\]
4. **Calculate the Probability**:
Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes:
\[
P(\text{only classical CDs}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{35}
\]
### Answer
Therefore, the probability that Clarice will select only classical CDs is \( \frac{1}{35} \).
This scenario demonstrates using combinatorics and probability to solve a real-life problem, which can help students understand the practical applications of these mathematical concepts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd89c40f4-4529-4a29-9a11-662a2483fc15%2Fdd6eb92b-0355-4716-8516-50dbc8c27139%2F82d6cc_processed.jpeg&w=3840&q=75)
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