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What is the Probability of Exactly 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 exactly one event happening means finding the chance that only one specific outcome occurs when you have multiple possibilities. It helps us calculate the likelihood of a single success among several trials. This is different from finding the chance of *at least one* event or *all* events happening.
Simple Example
Quick Example
Imagine you have two friends, Rahul and Priya, who both try to score a goal in a football match. The probability of exactly one event means finding the chance that *either* Rahul scores *or* Priya scores, but not both. It's like saying only one of them gets to celebrate!
Worked Example
Step-by-Step
Let's say a cricket team has two batsmen, Virat and Rohit. The probability of Virat scoring a boundary in the next ball is 0.6 (P(V)=0.6), and the probability of Rohit scoring a boundary in the next ball is 0.7 (P(R)=0.7). Assume their chances are independent. What is the probability that exactly one of them scores a boundary?
---Step 1: Understand the two scenarios for exactly one event. This means either Virat scores AND Rohit does NOT, OR Rohit scores AND Virat does NOT.
---Step 2: Calculate the probability of Virat scoring AND Rohit NOT scoring. P(Virat scores) = 0.6. P(Rohit does NOT score) = 1 - P(Rohit scores) = 1 - 0.7 = 0.3. So, P(V and not R) = P(V) * P(not R) = 0.6 * 0.3 = 0.18.
---Step 3: Calculate the probability of Rohit scoring AND Virat NOT scoring. P(Rohit scores) = 0.7. P(Virat does NOT score) = 1 - P(Virat scores) = 1 - 0.6 = 0.4. So, P(R and not V) = P(R) * P(not V) = 0.7 * 0.4 = 0.28.
---Step 4: Add the probabilities of these two mutually exclusive scenarios (since both cannot happen at the same time). P(exactly one) = P(V and not R) + P(R and not V) = 0.18 + 0.28 = 0.46.
---Answer: The probability that exactly one of them scores a boundary is 0.46.
Why It Matters
Understanding this concept is super important in many real-world fields. For example, in medicine, doctors use it to predict the chance of exactly one treatment working. In finance, it helps predict if exactly one stock will perform well, guiding investment decisions. Engineers use it to calculate the reliability of systems where only one component needs to succeed for a task.
Common Mistakes
MISTAKE: Adding the probabilities of individual events without considering the 'not happening' part. E.g., just adding P(A) + P(B). | CORRECTION: Remember that 'exactly one' means one happens AND the other does NOT. So, calculate P(A and not B) + P(B and not A).
MISTAKE: Confusing 'exactly one' with 'at least one'. 'At least one' means one or more events can happen. | CORRECTION: 'Exactly one' is very specific: only one event occurs. 'At least one' includes cases where both events might happen.
MISTAKE: Forgetting that events must be independent for direct multiplication of probabilities. | CORRECTION: If events are dependent, you need to use conditional probability (P(A and B) = P(A) * P(B|A)). For most basic 'exactly one' problems, independence is assumed or stated.
Practice Questions
Try It Yourself
QUESTION: Two students, Aman and Bani, are trying to solve a math problem. The probability that Aman solves it is 0.8, and the probability that Bani solves it is 0.5. What is the probability that exactly one of them solves the problem? | ANSWER: 0.5
QUESTION: A light bulb manufacturing unit produces two types of bulbs, A and B. The probability of bulb A being defective is 0.05, and for bulb B, it is 0.03. If you pick one of each type, what is the probability that exactly one of the bulbs is defective? | ANSWER: 0.077
QUESTION: In a game show, there are two doors. Behind Door 1, there's a prize with 0.6 probability. Behind Door 2, there's a prize with 0.7 probability. What is the probability that you win a prize from Door 1 but not Door 2, OR you win from Door 2 but not Door 1? | ANSWER: 0.46
MCQ
Quick Quiz
If P(A) = 0.4 and P(B) = 0.5, and A and B are independent events, what is the probability of exactly one of A or B occurring?
0.9
0.2
0.5
0.46
The Correct Answer Is:
C
P(exactly one) = P(A and not B) + P(B and not A). P(A and not B) = 0.4 * (1-0.5) = 0.4 * 0.5 = 0.2. P(B and not A) = 0.5 * (1-0.4) = 0.5 * 0.6 = 0.3. So, 0.2 + 0.3 = 0.5.
Real World Connection
In the Real World
In an app like Swiggy or Zomato, when you order food, there might be two delivery riders available. The system might calculate the probability that exactly one of them is available to pick up your order quickly, helping them assign the delivery efficiently. This ensures your hot food reaches you fast!
Key Vocabulary
Key Terms
PROBABILITY: The chance of an event happening, usually a number between 0 and 1. | INDEPENDENT EVENTS: Events where the outcome of one does not affect the outcome of the other. | MUTUALLY EXCLUSIVE: Events that cannot happen at the same time. | COMPLEMENTARY EVENT: The event that something does NOT happen (e.g., if P(A) is probability of A, then 1-P(A) is probability of not A).
What's Next
What to Learn Next
Great job understanding 'exactly one event'! Next, explore 'Probability of At Least One Event'. This builds on what you've learned and shows you how to handle scenarios where multiple successes are allowed, which is super useful for more complex problems.
