A local market sells avocados. Suppose there are 64 for sale and, unknown to you, 24 of them are rotten. If you were to purchase 4 avocados, chosen randomly from those available, let X denote the number of the purchased avocados that are rotten. What is the distribution of X? What is the probability that none of those purchased are rotten?
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.
![### Understanding the Probability Distribution in Sampling Avocados
**Problem Statement:**
A local market sells avocados. Suppose there are 64 for sale and, unknown to you, 24 of them are rotten. If you were to purchase 4 avocados, chosen randomly from those available, let \( X \) denote the number of the purchased avocados that are rotten.
- **Question 1:** What is the distribution of \( X \)?
- **Question 2:** What is the probability that none of those purchased are rotten?
**Explanation and Detailed Analysis:**
To determine the distribution of \( X \), consider that:
- The total number of avocados is 64.
- Out of these, 24 are rotten and 40 are not (64 - 24 = 40).
- You are randomly selecting 4 avocados.
Thus, \( X \) can take any value from 0 to 4, representing the possible number of rotten avocados in your sample.
To find the probability distribution of \( X \), we use the **Hypergeometric Distribution**. This distribution is appropriate when we are dealing with samples from a finite population without replacement. The hypergeometric distribution can be defined by three parameters:
1. **Population size (N)**: 64 (total avocados)
2. **Number of successes in population (K)**: 24 (rotten avocados)
3. **Sample size (n)**: 4 (avocados selected)
The probability mass function for the hypergeometric distribution is given as:
\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]
where:
- \( \binom{N}{n} \) represents the number of ways to choose \( n \) items from \( N \) items.
- \( k \) is the number of rotten avocados in your sample.
**Calculating the Probability that None are Rotten:**
To find the probability that none of the avocados you purchased are rotten (\( X = 0 \)), we substitute the values into the hypergeometric probability formula:
\[ P(X = 0) = \frac{\binom{24}{0} \binom{40}{4}}{\binom{64}{4}} \]
1. \( \binom{24}{0} \) is the number of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F091efdc9-e15b-4ccc-8207-a2051acfb2c9%2F058374a2-25af-4212-afa8-34f74cde9f77%2Fumbbt5f_processed.png&w=3840&q=75)
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