This problem presents a scenario involving the lifespan of Brand LT5 batteries. We're given that the median lifespan of these batteries is 100 hours. We need to calculate the probability that, in a set of three batteries, exactly two will last longer than 100 hours. This problem involves understanding basic probability...
Key Concepts: Binomial Probability
Binomial probability deals with situations where we have a fixed number of trials (in our case, three batteries), each trial has two possible outcomes (battery lasts longer than 100 hours or not), and the probability of success (lasting longer than 100 hours) remains constant across all trials.
Calculating the Probability
To solve this, we need to use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of exactly k successes in n trials.
- n is the number of trials (3 batteries).
- k is the number of successes (2 batteries lasting longer than 100 hours).
- p is the probability of success (probability of a battery lasting longer than 100 hours). Since the median lifespan is 100 hours, we assume a 50% chance of a battery lasting longer than 100 hours, so p = 0.5.
Applying the Formula
Let's apply the formula to our problem:
P(X = 2) = (3 choose 2) * 0.5^2 * (1-0.5)^(3-2)
Calculating the terms:
- (3 choose 2) = 3 (the number of ways to choose 2 batteries out of 3)
- 0.5^2 = 0.25
- (1-0.5)^(3-2) = 0.5
Therefore:
P(X = 2) = 3 * 0.25 * 0.5 = 0.375
Conclusion
The probability that exactly two out of three Brand LT5 batteries will last longer than 100 hours is 0.375 or 37.5%.
Additional Considerations
This calculation assumes that the lifespan of each battery is independent of the others, meaning the performance of one battery doesn't influence the others. In reality, there could be factors affecting the lifespan of batteries in a set, such as manufacturing variations, storage conditions, and usage patterns.
Furthermore, the median lifespan of 100 hours is a statistical measure. It's possible that the actual distribution of battery lifespans is not perfectly symmetrical around the median. In such cases, the probability of a battery lasting longer than 100 hours might not be exactly 50%.
Practical Implications
Understanding the probability of battery lifespan is essential for various applications, including:
- Product Design and Development: Manufacturers can use probability calculations to optimize battery designs and ensure that a certain percentage of batteries meet desired performance criteria.
- Quality Control: Quality control teams can utilize probability analysis to identify and address any anomalies in battery performance.
- Consumer Decisions: Consumers can use probability information to make informed decisions about battery purchases and understand the expected lifespan of their devices.
By understanding the concepts of binomial probability and applying them to real-world scenarios like battery lifespan, we gain valuable insights into the behavior of systems and can make more informed decisions.