# Probability: Given the Mean and Standard deviation

• Nikki10
In summary, the formula for calculating the probability given the mean and standard deviation is P(x) = (1/σ√(2π)) * e^(-(x-μ)^2 / 2σ^2), also known as the normal distribution formula. The probability value obtained from the formula represents the likelihood of a certain event occurring within a given range of values. It cannot be greater than 1, as a probability of 1 represents a 100% chance of an event occurring. The mean and standard deviation both affect the probability value, with higher values indicating a higher likelihood of the event occurring. While probability can provide insight, it cannot be used to predict future events with certainty.
Nikki10

## Homework Statement

The distribution of a drugs among addicts with autoimmune disorder is N(6.9, 16.81). What is the probability that one of these addicts enters your office and reports taking 16 drugs or more?

## The Attempt at a Solution

z = (16-6.9)/4.1 = 2.21
P(z >16) = 0.014 from the table
so P(z > 16) = 0.014 or 1.4%

Is this correct. It seems to simple and maybe I don't understand the concept

$$P[x\geq 16]=1-P[x < 16]$$

$$P[x\geq 16]=1-P[z < 2.21]= 1-0.986$$

thank you

## What is the formula for calculating probability given the mean and standard deviation?

The formula for calculating probability given the mean and standard deviation is:
P(x) = (1 / (σ√(2π))) * e^(-((x-μ)^2) / (2σ^2))
Where P(x) is the probability, σ is the standard deviation, μ is the mean, and e is the base of natural logarithm.

## How do you interpret the probability value?

The probability value represents the likelihood of a particular event occurring. A higher probability value means that the event is more likely to occur, while a lower probability value means that the event is less likely to occur.

## What is the relationship between mean and median in terms of probability distribution?

The mean and median are both measures of central tendency in a probability distribution. The mean represents the average value of all the data points, while the median represents the middle value when the data is arranged in ascending or descending order. In a symmetrical distribution, the mean and median will be the same. In a skewed distribution, the mean will be affected by extreme values, while the median will not be affected.

## How does the standard deviation affect the shape of a probability distribution?

The standard deviation measures the spread of data around the mean. A larger standard deviation indicates that the data points are more spread out, resulting in a wider and flatter probability distribution. A smaller standard deviation means that the data points are closer to the mean, resulting in a narrower and taller probability distribution.

## What is the significance of the Empirical Rule in probability distributions?

The Empirical Rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. This rule is significant because it allows us to make predictions about the likelihood of events occurring and to understand the spread of data in a probability distribution.

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