Question

Prove that \( 0 \leq P(E) \leq 1 \), where P(E) is the probability of the event E.

Hint:

This property is one of the core axioms of probability. Remember the two extremes: - \( P(E) = 0 \) means E is an impossible event. - \( P(E) = 1 \) means E is a certain event. All other probabilities fall between these two values.

Answer 1

Step 1: Understanding the Concept:
This is a fundamental axiom of probability theory. The probability of any event must be a value between 0 and 1, inclusive. We need to prove this using the basic definition of probability.
Step 2: Key Formula or Approach:
The classical definition of the probability of an event E is: \[ P(E) = \frac{\text{Number of favorable outcomes for E}}{\text{Total number of outcomes in the sample space}} = \frac{n(E)}{n(S)} \] We will use the properties of the sets E (event) and S (sample space).
Step 3: Detailed Explanation:
Let S be the sample space, which is the set of all possible outcomes of an experiment.
Let E be an event, which is a subset of the sample space (\( E \subseteq S \)).
Let n(S) be the total number of outcomes in S and n(E) be the number of outcomes in E.
Part 1: Proving \( P(E) \geq 0 \)
The number of outcomes in any event E, \( n(E) \), cannot be negative. The minimum possible number of outcomes is zero (for an impossible event). So, \( n(E) \geq 0 \).
The total number of outcomes in a sample space, \( n(S) \), must be positive for a non-trivial experiment, so \( n(S)>0 \).
Therefore, the ratio must be non-negative: \[ P(E) = \frac{n(E)}{n(S)} \geq \frac{0}{n(S)} \implies P(E) \geq 0 \] Part 2: Proving \( P(E) \leq 1 \)
Since the event E is a subset of the sample space S (\( E \subseteq S \)), the number of elements in E cannot exceed the number of elements in S. So, \( n(E) \leq n(S) \).
Dividing both sides of this inequality by the positive number \( n(S) \), we get: \[ \frac{n(E)}{n(S)} \leq \frac{n(S)}{n(S)} \] \[ P(E) \leq 1 \] Part 3: Combining the results
From Part 1 and Part 2, we have \( P(E) \geq 0 \) and \( P(E) \leq 1 \).
Combining these gives the final result: \[ 0 \leq P(E) \leq 1 \] Step 4: Final Answer:
By using the definitions of an event and sample space, we have proven that the probability of any event E must lie in the range [0, 1].