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Conditional probability helps us determine the likelihood of an event, given that another event has already occurred. In the context of the problem, imagine two components in a system where you know the lifetime of one. How would you calculate the lifetime of the other?An example is given in part a of the exercise: if the first component runs for exactly 3 months, what is the probability that the second runs for more than 2 months? To solve this, we use the conditional probability formula.* Formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] where \(A\) is the event we want to find the probability for, given \(B\), another event.
We adjust the joint probability density function (pdf) as per the fixed condition (here, the first component working for 3 months is fixed), and solve the integral accordingly. It involves integrating the adjusted pdf over the desired range to get the required probability.This concept is vital in predicting system performance based on known conditions, providing insight into dependent event probabilities.

A probability density function, or pdf, is a function that describes the likelihood for a continuous random variable to take on a particular value. For continuous variables, the probability of the exact value is technically zero, so we look at intervals and use the pdf to calculate probabilities over those intervals through integration.In the exercise, the joint pdf is \(f(x, y) = c[10-(x+y)]\). This represents how the likelihood of the lifetimes of both components (x and y) varies. * Steps to use a pdf: * Identify the pdf and its domain. * Integrate over the desired interval to find probabilities. * Ensure the total probability across the domain equals 1 by solving for any constants, as shown in the solution where \(c\) is found.
For our joint pdf, constraints are set: \(x > 0\), \(y > 0\), and \((x+y < 10)\). Solving the double integral of the pdf over this range gives the constant multiplier \(c\), ensuring the pdf correctly models the scenario. This is critical because it ensures the function does not overestimate or underestimate probabilities within its boundaries.Key takeaway: Understanding a pdf allows one to determine the distribution of probabilities for continuous random variables, essential for scenarios involving lifetimes, measurements, etc.

The binomial distribution comes into play when dealing with trials that have two possible outcomes, like success or failure, over a fixed number of trials. It's used here in part b of the question where we want to know the probability that at least half of 20 systems work for more than 3 months.Consider its properties:* Consists of \(n\) independent trials.* Each trial has exactly two outcomes.* \(p\) is the probability of success on a single trial.In this problem:* Each system is a trial.* The outcome is whether a system lasts more than 3 months (success).* Calculating the probability of at least 10 out of 20 systems succeeding involves using binomial probability: \[ P(X \geq 10) = \sum_{k=10}^{20} \binom{20}{k} p^k (1-p)^{20-k} \]
This sums the probabilities of getting 10 up to 20 systems to last more than 3 months.Understanding the binomial distribution is crucial in making decisions and predictions based on repeated independent trials with the same probability of outcome. It's applicable in diverse fields like quality control, finance, and even genetics!