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# A system consists of two identical pumps, $\# 1$ and $\# 2 .$ If one pump fails, the system will still operate. However, because of the added strain, the extra remaining pump is now more likely to fail than was originally the case. That is, $r=P(\# 2$ fails $| \# 1$ fails $)>P(\# 2$ fails $)=q .$ If at least one pump fails by the end of the pump design life in 7$\%$ of all systems and both pumps fail during that period in only $1 \%,$ what is the probability that pump $\# 1$ will fail during the pump design life?

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Probability Topics

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Simão A.

September 17, 2020

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### Video Transcript

all right, we have two identical pumps which are numbered number one and number two. We're given that the probability that both of them fail is 0.1 and the probability that either of them fail 0.7 and we're supposed to find the probability that one of them fails on its own. So they give us some variables up here, but quite honestly, we don't need them to solve this problem. Instead, I'm going to set some variables for some events. So let a equal the event at pump number one fails. And let's set b to be the event at pump number two fails. All right, well, we have our addition rule. So the probability of a union be because the probability of a plus the probability of B minus the probability of a Intersection B. So let's think about this in context. A Union B is the event that both are sorry. A Union B is the event that either of these happened either pump one fails or pumped to fails. Well, that's this over here. In addition, we have a intersection B, which means that they both fail at the same time remember, Union refers to just the addition of these two events. What happens if either of them happened or both happen while intersection is limited to exclusively what happens between those two events? So we have our minus 0.1 over here and keep in mind, Pump number one and pump number two are identical, so it's fair to assume that the probability they fail is the same. Therefore, since we need to find the probability that Pump one fails, we could just say Hey p of a equals p a V. So there we go. Now we do some algebra. We add 0.1 to both sides and then divide both sides by two. There we go. The probability of pump number one failing. It's 0.4 or 4%.

University of California - Los Angeles

#### Topics

Probability Topics