What is the Sum of All Probabilities = 1?

S1-SA4-0981

Grade Level:

Class 5

Maths, Data Science, AI, Statistics, Logic

Definition

What is it?

The 'Sum of All Probabilities = 1' means that if you list ALL possible things that can happen in a situation, and add up their chances (probabilities), the total will always be exactly 1. It tells us that one of these things MUST happen.

Simple Example

Quick Example

Imagine your favourite cricket team is playing. They can either Win, Lose, or Draw. If you add up the probability of them Winning, the probability of them Losing, and the probability of them Drawing, the total will be 1.

Worked Example

Step-by-Step

Let's say you have a bag with 5 red marbles and 5 blue marbles.

Step 1: Find the total number of marbles. Total = 5 (red) + 5 (blue) = 10 marbles.
---Step 2: Find the probability of picking a red marble. P(Red) = Number of red marbles / Total marbles = 5 / 10 = 1/2.
---Step 3: Find the probability of picking a blue marble. P(Blue) = Number of blue marbles / Total marbles = 5 / 10 = 1/2.
---Step 4: Add the probabilities of all possible outcomes. P(Red) + P(Blue) = 1/2 + 1/2.
---Step 5: Calculate the sum. 1/2 + 1/2 = 2/2 = 1.
Answer: The sum of all probabilities is 1.

Why It Matters

This rule is super important for understanding chances and predictions, used in everything from weather forecasting to making smart decisions in business. It helps Data Scientists predict trends and AI systems make logical choices, opening doors to careers in technology and research.

Common Mistakes

MISTAKE: Adding probabilities of events that are NOT all possible outcomes. | CORRECTION: Make sure you include every single possible thing that can happen in your calculation.

MISTAKE: Getting a sum greater than 1 or less than 1. | CORRECTION: Always double-check your calculations. If the sum isn't exactly 1, you've likely missed an outcome or made a mistake in division.

MISTAKE: Confusing probability (a fraction or decimal) with the number of outcomes. | CORRECTION: Probability is always a value between 0 and 1, representing the chance. The sum of these chances is 1, not the sum of the number of items.

Practice Questions

Try It Yourself

QUESTION: A spinner has 4 equal sections: Red, Blue, Green, Yellow. What is the probability of landing on Red? If you add the probabilities of landing on Red, Blue, Green, and Yellow, what will the sum be? | ANSWER: P(Red) = 1/4. The sum will be 1.

QUESTION: A weather report says there's a 0.6 (60%) chance of rain tomorrow. What is the probability that it will NOT rain tomorrow? | ANSWER: P(Not Rain) = 1 - P(Rain) = 1 - 0.6 = 0.4 (40%).

QUESTION: In a school election, Rahul has a 0.45 chance of winning, and Priya has a 0.30 chance. If only Rahul, Priya, and Sameer are running, what is Sameer's chance of winning? | ANSWER: P(Sameer) = 1 - (P(Rahul) + P(Priya)) = 1 - (0.45 + 0.30) = 1 - 0.75 = 0.25.

MCQ

Quick Quiz

If the probability of a bus arriving late is 0.25, what is the probability of the bus arriving on time?

0.25

0.75

The Correct Answer Is:

The bus can either be late or on time. These are the only two possibilities. So, P(Late) + P(On Time) must equal 1. Therefore, P(On Time) = 1 - P(Late) = 1 - 0.25 = 0.75.

Real World Connection

In the Real World

This concept is used by apps like Swiggy or Zomato to estimate delivery times. They calculate the probability of traffic, driver availability, and order complexity. By adding up all possible scenarios (on-time, slightly late, very late), they ensure the total probability equals 1, helping them give you accurate predictions for your food delivery.

Key Vocabulary

Key Terms

PROBABILITY: The chance of something happening, a number between 0 and 1 | OUTCOME: A possible result of an experiment or event | SUM: The total you get when you add numbers together | CERTAIN EVENT: An event that is sure to happen, with a probability of 1

What's Next

What to Learn Next

Next, you can explore 'Probability of Independent Events'. Understanding how individual probabilities add up to 1 will make it easier to calculate the chances of multiple things happening together.