Elementary Events & Probability Sum

In probability, an elementary event (also known as a simple event) is a single possible outcome of a random experiment. It’s the most basic outcome that can be observed. The set of all possible elementary events for a given experiment is called the sample space. A fundamental principle of probability states that when you consider all possible outcomes (elementary events) of an experiment, their probabilities must add up to 1 (or 100%). This is because one of these outcomes *must* occur.

Formulae

Let $E_1, E_2, …, E_n$ be the set of all elementary events in a sample space. Let $P(E_i)$ represent the probability of the elementary event $E_i$. Then:

$$\sum_{i=1}^{n} P(E_i) = P(E_1) + P(E_2) + … + P(E_n) = 1$$

This equation expresses that the sum of the probabilities of all possible elementary events is always equal to 1.

Examples

Example-1: Flipping a Fair Coin

Consider flipping a fair coin once. The elementary events are Heads (H) and Tails (T). Since the coin is fair, $P(H) = 0.5$ and $P(T) = 0.5$. The sum of the probabilities is $P(H) + P(T) = 0.5 + 0.5 = 1$.

Example-2: Rolling a Fair Six-Sided Die

When rolling a fair six-sided die, the elementary events are the numbers 1, 2, 3, 4, 5, and 6. The probability of each event is $\frac{1}{6}$ (assuming a fair die). The sum of the probabilities is: $P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = 1$.

Theorem with Proof

Theorem: The sum of the probabilities of all elementary events in a sample space is 1.

Proof:

Let $S$ be the sample space containing all possible outcomes of a random experiment. Let $E_1, E_2, …, E_n$ be the elementary events in $S$. By definition of probability, each $P(E_i)$ represents the probability of event $E_i$ occurring.

1. The Sample Space: The sample space $S$ encompasses *all* possible outcomes. This means that one and only one of the elementary events must occur when the experiment is performed.

2. Probability Axioms: We know that the probability of any event must be between 0 and 1 (inclusive). Also, we know that the sum of the probabilities of all possible outcomes must equal 1, which represents certainty. This is the formal basis of probability theory.

3. Certainty: Because the experiment *must* have one of these outcomes, the probability of the entire sample space, which is the union of all elementary events, is 1.

4. Mathematical Expression: Thus, $P(E_1 \cup E_2 \cup … \cup E_n) = P(E_1) + P(E_2) + … + P(E_n) = 1$. (We can add the probabilities directly because elementary events are mutually exclusive, meaning they cannot happen at the same time).

Therefore, the sum of the probabilities of all elementary events in a sample space is 1.

Common mistakes by students

• Not considering all outcomes: Students sometimes fail to identify *all* possible elementary events in a given experiment, leading to an incorrect calculation of probabilities.
• Incorrectly assigning probabilities: Students may incorrectly assign probabilities to elementary events. For example, they might think heads and tails on a fair coin have different probabilities.
• Confusing elementary events with other events: Students can confuse elementary events with compound events or other more complex event structures, which leads to error in the calculation.

Real Life Application

In weather forecasting, meteorologists use the sum of probabilities of different weather conditions to predict future weather. Let’s consider a single day: The elementary events could be: Sunny, Rainy, Cloudy. The probabilities might be, $P(Sunny) = 0.6$, $P(Rainy) = 0.3$, $P(Cloudy) = 0.1$. The sum $0.6 + 0.3 + 0.1 = 1$, indicating all possible weather scenarios for that day have been accounted for. This is useful for planning outdoor activities or making business decisions.

Fun Fact

The concept of probability was formally developed in the mid-17th century, largely through the correspondence between Blaise Pascal and Pierre de Fermat, while they were trying to solve gambling problems! Before that, people had intuitive notions of chance, but there wasn’t a standardized mathematical framework for it.

Recommended YouTube Videos for Deeper Understanding

Q.1 A fair six-sided die is rolled. What is the probability of rolling a number less than 3?

Check Solution

Ans: B

The elementary events are rolling a 1, 2, 3, 4, 5, or 6. The event of rolling a number less than 3 includes rolling a 1 or a 2. The probability is (favorable outcomes) / (total outcomes) = $2/6$.

Q.2 An urn contains 5 red balls and 3 blue balls. What is the sum of the probabilities of drawing a red ball and drawing a blue ball?

Check Solution

Ans: D

The elementary events are drawing a red ball or a blue ball. The probability of drawing a red ball is $5/8$. The probability of drawing a blue ball is $3/8$. The sum of these probabilities is $5/8 + 3/8 = 1$.

Q.3 A spinner has four equal sections colored red, blue, green, and yellow. What is the probability of the spinner landing on red or blue?

Check Solution

Ans: B

The elementary events are landing on red, blue, green, or yellow. The probability of red is $1/4$ and the probability of blue is $1/4$. The probability of red or blue is $1/4 + 1/4 = 2/4$.

Q.4 In a bag, there are only black and white marbles. The probability of picking a black marble is 0.35. What is the probability of picking a white marble?

Check Solution

Ans: B

The only elementary events are picking black or picking white. The sum of the probabilities must equal 1. Therefore, $P(\text{white}) = 1 – P(\text{black}) = 1 – 0.35 = 0.65$.

Q.5 A coin is flipped three times. What is the sum of the probabilities of all possible outcomes (e.g., HHH, HHT, HTH, etc.)?

Check Solution

Ans: D

The elementary events are all possible outcomes from flipping a coin three times. The sum of probabilities of all elementary events is always 1.

Next Topic: Sure & Impossible Events: Probability Values