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What is the Probability of at Least One Event?
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' means the chance that an event happens one or more times. It's often easier to calculate this by finding the probability that the event *never* happens and subtracting that from 1.
Simple Example
Quick Example
Imagine you have 3 chances to hit a six in a cricket match. What's the probability you hit 'at least one' six? Instead of calculating P(1 six) + P(2 sixes) + P(3 sixes), you can find P(no sixes) and subtract it from 1.
Worked Example
Step-by-Step
Suppose a factory produces LED bulbs, and there's a 10% chance (0.10) that any single bulb is defective. If you pick 2 bulbs, what is the probability that at least one of them is defective?
Step 1: Understand 'at least one defective'. This means either the first is defective, or the second is defective, or both are defective.
---Step 2: Find the probability that a single bulb is *not* defective. P(not defective) = 1 - P(defective) = 1 - 0.10 = 0.90.
---Step 3: Find the probability that *neither* bulb is defective. Since the choices are independent, multiply the probabilities: P(neither defective) = P(1st not defective) * P(2nd not defective) = 0.90 * 0.90 = 0.81.
---Step 4: Use the 'at least one' formula. P(at least one defective) = 1 - P(neither defective).
---Step 5: Calculate the final probability: P(at least one defective) = 1 - 0.81 = 0.19.
Answer: The probability that at least one of the two bulbs is defective is 0.19 or 19%.
Why It Matters
Understanding 'at least one' probability is super important in many fields! Engineers use it to predict if at least one part in a machine might fail. Doctors use it to understand the chance of at least one patient responding to a new medicine. Even in AI, it helps estimate the likelihood of at least one correct prediction.
Common Mistakes
MISTAKE: Adding probabilities directly when events are not mutually exclusive. For example, if P(A) is 0.2 and P(B) is 0.3, students might say P(at least one) = 0.2 + 0.3 = 0.5. | CORRECTION: This only works if events A and B cannot happen at the same time. Always use the 1 - P(none) method for 'at least one' to be safe.
MISTAKE: Forgetting that 'at least one' includes the possibility of *all* events happening. | CORRECTION: Remember, 'at least one' means 1, 2, 3... up to all events. The 1 - P(none) method correctly covers all these possibilities.
MISTAKE: Not correctly calculating P(none) when there are multiple trials. For example, if P(success) is 0.4, some might say P(no success in 3 trials) is 1 - 0.4 = 0.6. | CORRECTION: P(no success in 3 trials) is P(failure) * P(failure) * P(failure) = (1 - 0.4) * (1 - 0.4) * (1 - 0.4) = 0.6 * 0.6 * 0.6 = 0.216.
Practice Questions
Try It Yourself
QUESTION: A lucky draw has a 20% chance of winning a prize. If you buy one ticket, what is the probability of winning a prize? | ANSWER: 0.20
QUESTION: A lucky draw has a 20% chance of winning a prize. If you buy two tickets, what is the probability of winning at least one prize? | ANSWER: P(not winning) = 1 - 0.20 = 0.80. P(not winning both) = 0.80 * 0.80 = 0.64. P(at least one win) = 1 - 0.64 = 0.36.
QUESTION: A cricket player has a 0.3 probability of hitting a boundary (4 or 6) on any given ball. If they face 3 balls, what is the probability they hit at least one boundary? | ANSWER: P(no boundary) = 1 - 0.3 = 0.7. P(no boundary in 3 balls) = 0.7 * 0.7 * 0.7 = 0.343. P(at least one boundary) = 1 - 0.343 = 0.657.
MCQ
Quick Quiz
If the probability of getting a head on a single coin toss is 0.5, what is the probability of getting at least one head in two coin tosses?
0.25
0.5
0.75
1
The Correct Answer Is:
C
The probability of getting no heads (two tails) is 0.5 * 0.5 = 0.25. So, the probability of at least one head is 1 - 0.25 = 0.75.
Real World Connection
In the Real World
Imagine you're developing a new app like Paytm or PhonePe. Before launching, you want to know the probability that at least one user out of 100 will find a specific bug. Using this concept, you can calculate this chance. Or, if ISRO launches multiple satellites, they calculate the probability that at least one will successfully reach orbit, based on individual satellite success rates.
Key Vocabulary
Key Terms
PROBABILITY: The chance of an event happening, from 0 (impossible) to 1 (certain). | INDEPENDENT EVENTS: Events where the outcome of one does not affect the outcome of another. | COMPLEMENTARY EVENT: The event that something *doesn't* happen. P(A') = 1 - P(A). | AT LEAST ONE: Meaning one or more times.
What's Next
What to Learn Next
Now that you understand 'at least one' probability, you're ready to explore Conditional Probability! This concept helps you calculate probabilities when one event has already happened, which is super useful for making smarter predictions.
