Q.The sum of probabilities of two students getting distinction in their final examinations is $1.2$.

Concept: Axioms of Probability

The probability of any event must be a real number between $0$ and $1$, inclusive.

Let $P(S_1)$ be the probability of the first student getting distinction and $P(S_2)$ be the probability of the second student getting distinction.

By the axioms of probability, we have $0 \le P(S_1) \le 1$ and $0 \le P(S_2) \le 1$.

Therefore, the sum of their probabilities, $P(S_1) + P(S_2)$, can range from a minimum of $0+0=0$ to a maximum of $1+1=2$. …

Concept and Intuition: Understanding Probability Sums

Probability is a measure of the likelihood of an event occurring, always ranging from $0$ (impossible) to $1$ (certain). For any single event, its probability $P(E)$ must satisfy $0 \le P(E) \le 1$. This is a fundamental axiom of probability theory.

However, this rule applies to individual events. When we consider the sum of probabilities of multiple distinct events, say $P(A) + P(B)$, this sum is not necessarily bound by $1$. The sum can indeed be greater than $1$. This is because events $A$ and $B$ might not be mutually exclusive (they can both happen) and they are not necessarily exhaustive (they don't cover all possibilities).

For example, consider the probability of rain today ($P(R)$) and the probability of a sunny day tomorrow ($P(S)$). Both $P(R)$ and $P(S)$ must be between $0$ and $1$. If $P(R) = 0.8$ and $P(S) = 0.7$, then $P(R) + P(S) = 1.5$. This is a perfectly valid scenario, as long as $P(R)$ and $P(S)$ individually respect the $0 \le P(E) \le 1$ rule. The problem given is a direct application of this concept.

1. Define the Events and Given Information Let $P(D_1)$ be the probability that the first student gets a distinction in their final examinations. Let $P(D_2)$ be the probability that the second student gets a distinction in their final examinations. The problem states that the sum of these probabilities is $1.2$.

2. Recall the Fundamental Axiom of Probability For any event $E$, its probability $P(E)$ must satisfy:

This means that the probability of the first student getting a distinction must be between $0$ and $1$, and similarly for the second student:

3. Analyze the Sum of Probabilities The sum of probabilities of two separate events, $P(D_1) + P(D_2)$, is not constrained to be less than or equal to $1$. This constraint applies to the probability of a single event or the sum of probabilities of mutually exclusive and exhaustive events within a sample space. As long as the individual probabilities $P(D_1)$ and $P(D_2)$ are valid (i.e., between $0$ and $1$), their sum can exceed $1$. …