Geometric distributions model situations where you face repeated trials with a constant probability of success on each attempt. The key question often involves **cumulative probabilities**: what’s the likelihood of achieving success within a certain number of trials? Here’s how to tackle it:

**1. Understand the Players:**

**p:**Probability of success in each trial (must be between 0 and 1).**q:**Probability of failure (1 – p).**X:**Random variable representing the number of trials needed for the first success.

**2. The Formula:**

The **cumulative distribution function (CDF)** tells you the probability of obtaining success in **X ≤ x** trials:

```
P(X ≤ x) = 1 - (1 - p)^x
```

- This formula considers all outcomes
**up to and including**the xth trial.

**3. Interpreting the Result:**

- The value of P(X ≤ x) lies between 0 and 1.
- As x increases, P(X ≤ x) approaches 1 (certainty of success eventually).
- For a specific x, the result represents the chance of success within those x trials.

**4. Special Cases:**

**P(X = 1):**The probability of success on the very first trial, which equals p.**P(X > x):**The opposite of P(X ≤ x), calculated as 1 – P(X ≤ x).

**5. Beyond Formulas:**

- Calculators and statistical software can efficiently compute cumulative probabilities.
- Visualizing the geometric distribution through graphs can aid understanding.

**Remember:**

- The formula assumes independent trials with constant success probability.
- Adapt the approach if dealing with more complex scenarios involving additional parameters or dependencies.

In a geometric distribution, P(X > x) represents the probability that it takes **more than x trials** to achieve the first success. It’s essentially the opposite of the cumulative probability P(X ≤ x), which we discussed earlier.

Here’s how you can find P(X > x):

**1. Leverage the Relationship:**

P(X > x) is directly related to P(X ≤ x) through the following equation:

```
P(X > x) = 1 - P(X ≤ x)
```

**2. Apply the CDF Formula:**

Substitute the formula for P(X ≤ x) from the previous explanation:

```
P(X > x) = 1 - [1 - (1 - p)^x]
```

**3. Simplify:**

Combine the terms:

```
P(X > x) = (1 - p)^x
```

This formula directly gives you the probability of needing more than x trials for success in the geometric distribution.

**Example:**

Suppose the probability of success in each trial is p = 0.4 and you’re interested in P(X > 3).

```
P(X > 3) = (1 - 0.4)^3 = 0.216
```

Therefore, there’s a 38.4% chance that it will take more than 3 trials for the first success to occur in this scenario.