What is Probability of At Least One Event? | Simple Explanation for Class 12

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What is the Probability of at Least One Event (General Case)?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The probability of 'at least one' event happening means the chance that one or more of several possible outcomes will occur. It's often easier to calculate this by finding the probability that *none* of the events happen and subtracting that from 1.

Simple Example

Quick Example

Imagine you have 3 friends, and each friend might bring a samosa to your party. The probability of 'at least one samosa' means the chance that you get 1 samosa, or 2 samosas, or even all 3 samosas. It's easier to think: what's the chance *no one* brings a samosa, and then subtract that from 1.

Worked Example

Step-by-Step

PROBLEM: You are taking two online quizzes, Quiz A and Quiz B. The probability of passing Quiz A is 0.8, and the probability of passing Quiz B is 0.7. What is the probability of passing at least one quiz, assuming the quizzes are independent?

Step 1: Understand 'at least one'. This means passing Quiz A, or passing Quiz B, or passing both.
---Step 2: Find the probability of NOT passing each quiz. If P(Pass A) = 0.8, then P(Fail A) = 1 - 0.8 = 0.2. If P(Pass B) = 0.7, then P(Fail B) = 1 - 0.7 = 0.3.
---Step 3: Find the probability of NOT passing *any* quiz (i.e., failing both). Since quizzes are independent, P(Fail A AND Fail B) = P(Fail A) * P(Fail B) = 0.2 * 0.3 = 0.06.
---Step 4: The probability of passing at least one quiz is 1 minus the probability of failing both. So, P(at least one pass) = 1 - P(Fail A AND Fail B) = 1 - 0.06.
---Step 5: Calculate the final answer. 1 - 0.06 = 0.94.

ANSWER: The probability of passing at least one quiz is 0.94.

Why It Matters

Understanding 'at least one' probability is super useful! In medicine, doctors use it to see the chance of a patient responding to at least one treatment. In AI/ML, it helps predict if at least one model will identify a pattern. It's key for engineers designing systems where multiple parts could fail but the system still works if at least one part is fine.

Common Mistakes

MISTAKE: Adding probabilities directly for 'at least one' without considering overlap. For example, P(A) + P(B) for P(A or B). | CORRECTION: Use the formula P(A or B) = P(A) + P(B) - P(A and B) for non-mutually exclusive events, or the complement rule: 1 - P(none of them).

MISTAKE: Forgetting that 'at least one' includes the possibility of ALL events happening. | CORRECTION: Remember that 'at least one' means one, or two, or three, up to all possible events occurring. The complement rule (1 - P(none)) naturally covers all these cases.

MISTAKE: Not correctly calculating the probability of 'none' happening, especially when events are dependent. | CORRECTION: Always ensure you correctly find the probability of each event NOT happening, and then multiply them if events are independent. If dependent, the calculation for 'none' becomes more complex and needs conditional probability.

Practice Questions

Try It Yourself

QUESTION: A fair coin is tossed two times. What is the probability of getting at least one head? | ANSWER: 0.75

QUESTION: A factory produces LED bulbs. The probability that a bulb is defective is 0.05. If two bulbs are chosen randomly, what is the probability that at least one of them is defective? | ANSWER: 0.0975

QUESTION: In a cricket match, Virat Kohli has a 0.6 probability of scoring more than 50 runs, and Rohit Sharma has a 0.7 probability of scoring more than 50 runs. Assuming their performances are independent, what is the probability that at least one of them scores more than 50 runs? | ANSWER: 0.88

MCQ

Quick Quiz

If the probability of rain on Monday is 0.4 and on Tuesday is 0.5 (independent events), what is the probability of rain on at least one of these days?

0.2

0.7

0.9

0.8

The Correct Answer Is:

P(no rain Monday) = 1 - 0.4 = 0.6. P(no rain Tuesday) = 1 - 0.5 = 0.5. P(no rain on both days) = 0.6 * 0.5 = 0.30. P(at least one day of rain) = 1 - 0.30 = 0.70.

Real World Connection

In the Real World

Imagine you're ordering food using a delivery app like Swiggy or Zomato. The app might estimate the probability of your order arriving late from two different restaurants. The concept of 'at least one' helps them calculate the chance that *either* one or both orders might be late, allowing them to manage customer expectations or offer discounts proactively.

Key Vocabulary

Key Terms

PROBABILITY: The chance of an event happening | COMPLEMENT: The event not happening; if P(A) is probability of A, P(not A) = 1 - P(A) | INDEPENDENT EVENTS: Events where the outcome of one does not affect the outcome of the other | AT LEAST ONE: One or more of the events occur

What's Next

What to Learn Next

Next, you can explore 'Conditional Probability' and 'Bayes' Theorem'. These build on understanding how individual probabilities combine and change when you have more information, which is crucial for advanced topics in AI and data science.