Consider the experiment of tossing a pair of dice and recording the results as "high" if the die displays a value of 4,5, or 6 and"low" if it displays a value of 1,2,0r 3. Define the events A ="the first die falls high", B="their total is seven", and C ="the total of the values displayed is four". Determine whether A and B are independent.
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: Independence of Events
Consider the experiment of tossing a pair of dice and recording the results as “high” if the die displays a value of 4, 5, or 6 and “low” if it displays a value of 1, 2, or 3. Define the events \( A \) as “the first die falls high”, \( B \) as “their total is seven”, and \( C \) as “the total of the values displayed is four”.
1. **Determine whether \( A \) and \( B \) are independent.**
2. **Determine whether \( A \) and \( C \) are independent.**
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### Explanation:
In order to solve the problems, we need to determine the independence of events \( A \) and \( B \), and \( A \) and \( C \). The events are defined as follows:
- \( A \): The first die shows a high value (4, 5, or 6).
- \( B \): The sum of the values on the two dice is 7.
- \( C \): The sum of the values on the two dice is 4.
For two events \(X\) and \(Y\) to be independent, the probability of both events happening simultaneously must be equal to the product of their individual probabilities:
\[ P(X \cap Y) = P(X) \cdot P(Y) \]
We will follow these steps for each pair of events \( (A, B) \) and \( (A, C) \):
1. Calculate \( P(A) \), \( P(B) \), and \( P(C) \).
2. Calculate \( P(A \cap B) \) and \( P(A \cap C) \).
3. Check if \( P(A \cap B) = P(A) \cdot P(B) \) and if \( P(A \cap C) = P(A) \cdot P(C) \).
Finally, deduce whether \( A \) and \( B \) are independent, and whether \( A \) and \( C \) are independent based on these calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00a48d79-f805-418c-a679-1c91878d1d75%2F5ec9de66-320d-442b-b7b6-1d8113146538%2F9kewal_processed.png&w=3840&q=75)
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