23) 84% of a contractor's jobs involves electrical work. 75% of a contractor's jobs involve plumbing work. Of the jobs that involve plumbing, 90% of the jobs also involves electrical work. Let E = jobs involving electrical work L = jobs involving plumbing work Suppose one of the contractor's jobs is randomly selected. Using the sixth Excel worksheet, a) Find P(E). 0.84 b) Find P(L). 0.75 c) In words, what does E | L mean? d) Find P(E | L). e) Find P(E and L). f) Are E and L independent events? Why or why not? g) Find P(E or L). h) Are E and L mutually exclusive? Why or why not

Part a)

You are correct. P(E) = 0.84

Part b)

You are correct. P(L) = 0.75

Part c)

E | L means "jobs involving electrical work given a job involving plumbing"

It probably seems more familiar to have it as P(E | L) which means "the probability of getting electrical work given you got a job involving plumbing"

Part d)

P(E | L) = 0.90

Why is this? This is given in the statement " Of the jobs that involve plumbing, 90% of the jobs also involves electrical work". This is basically saying, if we know for certain we got a plumbing job, then there's a 90% chance we have electrical work as well.

In other words, we basically are saying P(E) = 0.90 given event L has already happened. We cannot just simply say P(E) = 0.90 without the "given event L" because this would contradict part (a) above.

Note how this value is different from P(E) = 0.84, which will come in handy for part (f).

Part e)

P(E | L) = P(E and L)/P(L) ... conditional probability formula

P(E | L)*P(L) = P(E and L) ... multiply both sides by P(L)

P(E and L) = P(E|L)*P(L) ... swap the two sides

P(E and L) = 0.90*0.75 plug in given values

P(E and L) = 0.675 ... is the answer

We will use this for comparison purposes in part (f), and we will also use it in part (g).

Part f)

If events E and L were independent, then this equation below would be true

P(E and L) = P(E)*P(L)

Let's see what we get when we multiply P(E) and P(L)

P(E)*P(L) = 0.84*0.75

P(E)*P(L) = 0.63

This is not the same value we got back in part (e), therefore,

P(E and L) = P(E)*P(L) is a false equation

So E and L are not independent events

We say E and L are dependent events. One event depends on the other, or vice versa (or perhaps both are linked in some way). The prior knowledge that we have plumbing work affects the probability of electrical work.

We can see this by comparing the values from parts (a) and (d). If E and L were independent, then P(E) = P(E | L) should be true, but it is not true.

Part g)

Use the values from parts (a), (b) and (e) to get

P(E or L) = P(E) + P(L) - P(E and L)

P(E or L) = 0.84 + 0.75 - 0.675

P(E or L) = 0.915

Part h)

Because the result from part (g) was not zero, this means E and L are not mutually exclusive.

Mutually exclusive events are two or more events that cannot happen at the same time. For example, it is impossible to get heads and tails at the same time on the same coin, so P(H or T) = 0 where H = heads and T = tails. For this example, we say the events are mutually exclusive. The number 0 in probability terms means impossibility, or that there's a 0% chance of it happening.

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