Two pumps connected in parallel fail independently of each other on any given day. The probabili

Two pumps connected in parallel fail independently of each...

Key Concepts

Complementary Events

Complementary events are pairs of outcomes that account for all possibilities, such that the sum of their probabilities equals one. In reliability and failure analysis, knowing the probability that an event does not occur can help in determining the probability that it does occur, especially when only partial information is provided about various outcomes.

System of Probability Equations

Often in probability problems, information is provided indirectly through 'only' or 'exactly one' type data, which leads to setting up systems of equations. By expressing the probabilities of compound events in terms of individual event probabilities and using their relationships, one can solve for unknown parameters. This technique is especially useful in problems where individual probabilities are not given directly but can be deduced from related outcomes.

Independent Events

This concept means that the occurrence of one event does not affect the occurrence of another. In probability problems involving multiple independent events, the probability of their joint occurrence is the product of their individual probabilities. Understanding independence is essential when calculating the probability of simultaneous failures or successes in separate components or trials.

Joint Probability

Joint probability refers to the likelihood of two or more events happening at the same time. When events are independent, the joint probability is calculated by multiplying the probability of each event. This concept is critical when determining the chance that multiple independent components or occurrences will all fail or succeed concurrently.

Transcript

00:02 So in this problem, we need to find the probability that both of these pumps fail, the old and the new.

00:12 So let's write down some things that we know.

00:14 So let's suppose p is the probability that the old pump fails.

00:20 And q is the probability that the new fails.

00:27 So then 1 minus p would be the old pump working.

00:34 So this is the old one works.

00:39 And 1 minus q would be the chances of the new one working.

00:46 And we know these things are independent of each other because that's given in the question.

00:51 So what we want to find here is the probability that they both fail.

00:56 So we want to figure out what is the probability p times q.

01:06 But this is a little tricky to find.

01:08 So this problem seems kind of easy at first glance, but there's really a lot.

01:11 A lot behind it.

01:14 So what do we know in the question? the question tells us that the probability that only the old one fails is 10%.

01:27 So what does that mean? if only the old one fails, that means the old fails, so we have p, and the new one works.

01:37 So that's 1 minus q.

01:39 And we know that that is equal to 10%.

01:44 So 0 .10.

01:47 We also know that the probability that only the new one fails, so what would that look like? we have q, new fails, but that means that the old one has to work, right? so 1 minus p, we know that that's equal to 5%.

02:05 So using these two equations, we need to somehow figure out what is p times q.

02:12 So how can we do that? well, let's do some work to these equations.

02:20 So first, let's call this equation one, equation two.

02:26 So let's do some work to equation one.

02:28 Let's get an expression for p.

02:32 So equation one, p is equal to 0 .10 over 1 minus q.

02:42 And let's plug this into equation 2.

02:45 So equation 2, we have q times 1 minus p.

02:51 So i'm going to plug in this expression for p is equal to 0 .05.

02:59 All right.

03:00 So let's simplify this a little bit.

03:04 Let's expand the parentheses.