Let X be a discrete random variable. If Pr(X<10) = 1/4, and Pr(X>10) = 1/5, then **what is Pr(X=10)?**

Please specify your answer in decimal terms and round your answer to the nearest hundredth (e.g., enter 12 percent as 0.12).

We are given that \( X \) is a discrete random variable, and the following probabilities are provided:

- \( \Pr(X < 10) = \frac{1}{4} \)
- \( \Pr(X > 10) = \frac{1}{5} \)

We need to find \( \Pr(X = 10) \).

Since \( X \) can either be less than 10, greater than 10, or equal to 10, we can use the following relationship for probabilities:

\[
\Pr(X < 10) + \Pr(X = 10) + \Pr(X > 10) = 1
\]

Substitute the given values:

\[
\frac{1}{4} + \Pr(X = 10) + \frac{1}{5} = 1
\]

Convert the fractions to have a common denominator. The least common denominator of 4 and 5 is 20, so:

\[
\frac{1}{4} = \frac{5}{20} \quad \text{and} \quad \frac{1}{5} = \frac{4}{20}
\]

Now substitute these into the equation:

\[
\frac{5}{20} + \Pr(X = 10) + \frac{4}{20} = 1
\]

Simplify the fractions:

\[
\frac{9}{20} + \Pr(X = 10) = 1
\]

Now, solve for \( \Pr(X = 10) \):

\[
\Pr(X = 10) = 1 - \frac{9}{20} = \frac{11}{20}
\]

Convert \( \frac{11}{20} \) to decimal form:

\[
\Pr(X = 10) = 0.55
\]

Thus, the probability \( \Pr(X = 10) \) is **0.55**.

Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:
1. What is the sum of probabilities of all possible outcomes for a random variable?
2. How does the probability distribution for a discrete random variable differ from that of a continuous one?
3. If \( \Pr(X < 10) \) had been \( \frac{1}{3} \), how would that affect \( \Pr(X = 10) \)?
4. Can you derive the cumulative distribution function (CDF) from given probabilities?
5. How would the answer change if \( X \) were continuous instead of discrete?

**Tip:** Always ensure the total probability across all events sums to 1 when working with probabilities.

Let X be a discrete random variable. If Pr(X < 10) = 1/4, and Pr(X > 10) = 1/5, then what is Pr(X = 10)? Please specify your answer in decimal terms and round your answer to the nearest hundredth (e.g., enter 12 percent as 0.12).

This problem explores how to find Pr(X = 10) for a discrete random variable given that Pr(X < 10) = 1/4 and Pr(X > 10) = 1/5. We solve using the total probability theorem and convert fractions to decimals. Suitable for high school students studying probability.

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