For a continuous random variable X, P(25 ≤ X ≤ 73) = 0.28 and P(X > 73) = 0.12. Calculate the following probabilities.

Note: Round your answers to 2 decimal places.

﻿﻿P(X < 73)
	﻿﻿﻿P(X < 25)
	﻿﻿﻿P(X = 73

We are given the following probabilities for the continuous random variable \( X \):

- \( P(25 \leq X \leq 73) = 0.28 \)
- \( P(X > 73) = 0.12 \)

### Step-by-step solution:

1. **Calculate \( P(X < 73) \)**:
   The total probability for any continuous random variable is 1. Therefore:
   \[
   P(X < 73) = 1 - P(X > 73)
   \]
   Substituting the given probability:
   \[
   P(X < 73) = 1 - 0.12 = 0.88
   \]

2. **Calculate \( P(X < 25) \)**:
   We can break down the probabilities around 25 and 73 as follows:
   \[
   P(25 \leq X \leq 73) + P(X < 25) + P(X > 73) = 1
   \]
   Rearranging this:
   \[
   P(X < 25) = 1 - P(25 \leq X \leq 73) - P(X > 73)
   \]
   Substituting the known values:
   \[
   P(X < 25) = 1 - 0.28 - 0.12 = 0.60
   \]

3. **Calculate \( P(X = 73) \)**:
   Since \( X \) is a continuous random variable, the probability of it taking any exact value is 0. Therefore:
   \[
   P(X = 73) = 0
   \]

### Final answers:
- \( P(X < 73) = 0.88 \)
- \( P(X < 25) = 0.60 \)
- \( P(X = 73) = 0 \)

Would you like further details or have any questions?

#### Related Questions:
1. How would you calculate \( P(X > 25) \)?
2. What is the cumulative distribution function (CDF) based on this data?
3. How do you find the probability if the distribution was discrete instead of continuous?
4. What if \( P(X > 25) \) was given, how would that change the calculations?
5. How can you use this information to calculate the mean or variance of \( X \)?

**Tip**: For continuous random variables, the probability of an exact value is always zero. This is a key property that distinguishes continuous distributions from discrete ones.

For a continuous random variable X, P(25 ≤ X ≤ 73) = 0.28 and P(X > 73) = 0.12. Calculate the following probabilities:
P(X < 73), P(X < 25), and P(X = 73).

Learn how to calculate probabilities for a continuous random variable X where P(25 ≤ X ≤ 73) = 0.28 and P(X > 73) = 0.12. Step-by-step solution includes calculating P(X < 73), P(X < 25), and understanding why P(X = 73) equals zero. Suitable for Grades 11-12 and early college students.

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