Are event A and event B independent? Justify your answer numerically (be exact here!).
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.
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- Are
event A and event B independent? Justify your answer numerically (be exact here!).
![# Standard Normal Distribution
This image illustrates a standard normal distribution curve along with a z-table.
### Formula:
The probability density function for the standard normal distribution is given by:
\[
p(z \leq z_1) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z_1} e^{-\frac{1}{2}z^2} \, dz
\]
### Z-Table Explanation:
The table below the curve is a standard normal (z) table, which provides the cumulative probabilities associated with each z-score in a standard normal distribution.
- **Left Column (`z_1`)**: This column lists the z-scores from 0.0 to 3.9, incrementing by 0.1.
- **Row & Columns (0.00 to 0.09)**: These represent the hundredths place of the z-score.
### How to Read the Table:
1. Locate the decimal value of the z-score (e.g., 2.1).
2. Move across to the column reflecting the hundredths value (e.g., 0.03).
3. The intersection gives the cumulative probability. For z = 2.13, the probability is 0.9834 (highlighted).
The highlighted rows and columns demonstrate notable cumulative probabilities at specific z-scores. For example:
- z = 0.05 corresponds to a probability of approximately 0.5199.
- z = 2.1 corresponds to a probability of approximately 0.9830.
This table serves as an essential tool for statistical calculations in determining the likelihood that a standard normal random variable is less than or equal to a given z-score.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f6bd7ce-0958-48a2-a65d-e948a47e92ca%2F768dd01a-052d-42fc-b81e-e016b32f17ee%2Fr9pvcok_processed.png&w=3840&q=75)
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