What is P(E) + P(not E) = 1?

S3-SA3-0070

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

P(E) + P(not E) = 1 means that the probability of an event happening (P(E)) plus the probability of that event NOT happening (P(not E)) always adds up to 1. This '1' represents 100% certainty, meaning something will either happen or not happen, there's no other option.

Simple Example

Quick Example

Imagine you're watching an India vs. Pakistan cricket match. India will either win or not win (which means they lose or draw). The probability of India winning plus the probability of India not winning will always be 1. You can't have any other outcome!

Worked Example

Step-by-Step

Let's say the probability of rain tomorrow in Bengaluru is 0.4 (P(Rain) = 0.4).
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We want to find the probability that it will NOT rain tomorrow (P(not Rain)).
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Using the formula: P(Rain) + P(not Rain) = 1
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Substitute the known value: 0.4 + P(not Rain) = 1
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To find P(not Rain), subtract 0.4 from 1: P(not Rain) = 1 - 0.4
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P(not Rain) = 0.6
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So, the probability that it will not rain tomorrow in Bengaluru is 0.6.

Why It Matters

This fundamental rule is super important in fields like Data Science and AI, helping predict outcomes from weather forecasts to stock market trends. It's used by engineers to design safe systems and by economists to understand market risks, making it a key tool for many future careers.

Common Mistakes

MISTAKE: Forgetting that 'not E' includes all other possibilities, not just one specific opposite. | CORRECTION: 'not E' means any outcome where E does not happen. For example, if E is 'getting a 6 on a die', 'not E' is 'getting 1, 2, 3, 4, or 5'.

MISTAKE: Expressing probability as a percentage and then adding it directly to a decimal. | CORRECTION: Always convert percentages to decimals (divide by 100) or decimals to percentages before adding, ensuring both are in the same format to sum to 1 or 100%.

MISTAKE: Assuming P(E) + P(not E) = 1 only applies to 'either/or' situations with two clear outcomes. | CORRECTION: This rule applies universally to any event 'E' and its complement 'not E', no matter how many other possible outcomes there are, as long as 'not E' covers all of them.

Practice Questions

Try It Yourself

QUESTION: The probability of your school bus arriving on time is 0.85. What is the probability that it will NOT arrive on time? | ANSWER: 0.15

QUESTION: A student has a 60% chance of passing their Maths exam. What is the probability (as a decimal) that they will NOT pass the exam? | ANSWER: 0.4

QUESTION: In a box of 50 pens, 10 are blue, 15 are red, and 25 are black. If the probability of picking a blue pen is 0.2, what is the probability of NOT picking a blue pen? | ANSWER: 0.8

MCQ

Quick Quiz

If the probability of a new smartphone model being successful is 0.7, what is the probability of it NOT being successful?

0.7

0.3

The Correct Answer Is:

The sum of the probability of an event happening and not happening must be 1. So, P(success) + P(not success) = 1. If P(success) = 0.7, then P(not success) = 1 - 0.7 = 0.3.

Real World Connection

In the Real World

When a food delivery app like Swiggy or Zomato predicts your order arrival time, they use probability. They calculate the chance of delivery being on time (P(on time)) and the chance of it being late (P(not on time)). This helps them manage customer expectations and driver routes more efficiently.

Key Vocabulary

Key Terms

PROBABILITY: The chance of an event happening, expressed as a number between 0 and 1 | EVENT: A specific outcome or set of outcomes in an experiment | COMPLEMENTARY EVENTS: Two events where one event happening means the other cannot, and together they cover all possibilities | CERTAINTY: A probability of 1, meaning an event is sure to happen

What's Next

What to Learn Next

Now that you understand P(E) + P(not E) = 1, you're ready to explore how to calculate probabilities for multiple events using concepts like 'AND' and 'OR'. This will help you solve even more complex probability problems!