In Example 99, the probability that `X ge 3` equals

Options

A. `(125)/(216)`
B. `(25)/(36)`
C. `(5)/(36)`
D. `(25)/(216)`

Explanation

To solve the problem of finding the probability that \( X \geq 3 \), we can follow these steps:

### Step 1: Understand the Probability Distribution
We need to identify the probability distribution of the random variable \( X \). In many cases, \( X \) could represent the outcome of a die roll or a similar discrete random variable.

### Step 2: Calculate the Probability of \( X < 3 \)
To find \( P(X \geq 3) \), we can first find \( P(X < 3) \) and then use the complement rule:
\[ P(X \geq 3) = 1 - P(X < 3) \]

### Step 3: Determine \( P(X < 3) \)
If \( X \) represents the outcomes of rolling a fair six-sided die, then:
- \( P(X = 1) = \frac{1}{6} \)
- \( P(X = 2) = \frac{1}{6} \)

Thus, we can calculate:
\[ P(X < 3) = P(X = 1) + P(X = 2) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \]

### Step 4: Calculate \( P(X \geq 3) \)
Now we can substitute \( P(X < 3) \) into our earlier equation:
\[ P(X \geq 3) = 1 - P(X < 3) = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 5: Conclusion
Thus, the probability that \( X \geq 3 \) is:
\[ P(X \geq 3) = \frac{2}{3} \]

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