In a probability distribution of a random variable X , the sum of probabilities is always

Options

A. a) 0
B. b) 1
C. c) 2
D. d) a non-negative integer

Answer

To solve the question regarding the sum of probabilities in a probability distribution of a random variable \( X \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Probability Distribution**:
A probability distribution assigns probabilities to each possible outcome of a random variable. The random variable can take on various values, and each value has an associated probability.

**Hint**: Remember that a probability distribution is a way to describe how probabilities are assigned to different outcomes.

2. **Definition of Probability**:
The probability of an event is defined as the number of favorable outcomes divided by the total number of possible outcomes. For a random variable, this means that the probabilities assigned to all possible outcomes must add up to a specific value.

**Hint**: Recall the formula for probability: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).

3. **Total Probability**:
In any probability distribution, the sum of the probabilities of all possible outcomes must equal 1. This is because the total probability represents the certainty that one of the possible outcomes will occur.

**Hint**: Think about the concept of certainty in probability; if you consider all possible outcomes, one of them must happen.

4. **Example with Coin Toss**:
Consider a simple example of tossing a fair coin. The possible outcomes are heads (H) and tails (T). The probabilities are:
- \( P(H) = \frac{1}{2} \)
- \( P(T) = \frac{1}{2} \)

When we add these probabilities together:
\[ P(H) + P(T) = \frac{1}{2} + \frac{1}{2} = 1 \]

**Hint**: Use simple examples like coin tosses or dice rolls to visualize how probabilities add up.

5. **Conclusion**:
Therefore, in any probability distribution of a random variable \( X \), the sum of all probabilities must always equal 1.

**Final Answer**: The correct answer to the question is **1**.