Normal Distribution: Consider a normally distributed random variable with mean 0 and standard deviation 1. How does the probability that 
 fall within the interval  [−1,1]  relate to probability that 
 fall within the interval  [0,1]  ?

For a normally distributed random variable \( X \) with mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \), the probability that \( X \) falls within certain intervals can be related to the properties of the **standard normal distribution** (denoted as \( Z \sim N(0, 1) \)).

### 1. Probability that \( X \) falls within the interval \([-1, 1]\):

We need to compute the probability \( P(-1 \leq X \leq 1) \), which is the area under the standard normal curve from -1 to 1:

\[
P(-1 \leq X \leq 1) = P(Z \leq 1) - P(Z \leq -1)
\]

Using the cumulative distribution function (CDF) for a standard normal distribution, we have:

\[
P(Z \leq 1) \approx 0.8413 \quad \text{and} \quad P(Z \leq -1) \approx 0.1587
\]

So:

\[
P(-1 \leq Z \leq 1) = 0.8413 - 0.1587 = 0.6826
\]

This means approximately **68.26%** of the values of a standard normal variable fall within one standard deviation of the mean (between -1 and 1).

### 2. Probability that \( X \) falls within the interval \([0, 1]\):

Now, we compute \( P(0 \leq X \leq 1) \), which is the area under the standard normal curve from 0 to 1:

\[
P(0 \leq Z \leq 1) = P(Z \leq 1) - P(Z \leq 0)
\]

Since \( P(Z \leq 1) \approx 0.8413 \) and \( P(Z \leq 0) = 0.5 \), we have:

\[
P(0 \leq Z \leq 1) = 0.8413 - 0.5 = 0.3413
\]

Thus, approximately **34.13%** of the values of a standard normal variable fall between 0 and 1.

### 3. Relationship Between the Two Probabilities:

The probability of falling within \([-1, 1]\) is **68.26%**, while the probability of falling within \([0, 1]\) is **34.13%**. Since the standard normal distribution is symmetric around zero, the probability that \( Z \) falls between -1 and 0 is also 34.13%, meaning that:

\[
P(-1 \leq Z \leq 1) = 2 \times P(0 \leq Z \leq 1)
\]

In other words, the probability of \( Z \) falling within \([-1, 1]\) is exactly **twice** the probability of \( Z \) falling within \([0, 1]\).

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Would you like any further details or explanations?

Here are five related questions to explore:
1. What is the probability of a standard normal variable falling between -2 and 2?
2. How does the probability change if the interval is expanded to \([-2, 2]\)?
3. Can you derive the probability for intervals such as \([1, 2]\) using the CDF?
4. What are the properties of the normal distribution that lead to symmetry?
5. How does the concept of z-scores relate to probabilities in normal distribution?

**Tip**: Always remember that about 68.26%, 95.44%, and 99.73% of the data falls within 1, 2, and 3 standard deviations of the mean, respectively, in a standard normal distribution.

Normal Distribution: Consider a normally distributed random variable with mean 0 and standard deviation 1. How does the probability that it falls within the interval [-1,1] relate to the probability that it falls within the interval [0,1]?

Explore how the probability of a standard normal variable falling within [-1, 1] compares to the probability of it falling within [0, 1]. Learn key concepts of the normal distribution, including the 68-95-99.7 rule and symmetry properties. Suitable for Grades 10-12.

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