Question

Jason inherited a piece of land from his great-uncle. Owners in the area claim that there is a 45% chance that the land has oil. Jason decides to test the land for oil. He buys a kit that claims to have an 80% accuracy rate of indicating oil in the soil. If the test predicts that there is no oil, what is the probability after the test that the land has oil?

Answer 1

0.1698

0.2217

0.5532

0.7660

Answer 2

@YanaSidlinskiy

Answer 3

Again, let
A=the land has oil
B=the test says the land has oil
we then want P(A|B)

Answer 4

What is the formula for P(A!B)

Answer 5

\[P(A|B)=\frac{P(A\cap B)}{P(B)}\]

Answer 6

Is that a*b/b?

Answer 7

well you can't multiply events like that, but no it is not the same as P(A)*P(B)/P(B)=P(A)
because
\(P(A\cap B)=P(A)P(B)\) if and only if A and B are independent, which they are not in this case since getting a positive test would change our expectation of finding oil

Answer 8

so again you will need to draw a tree

Answer 9

so its c

Answer 10

can you draw the tree

Answer 11

I don't know I haven't done the problem yet

Answer 12

well first we will need to draw the fork for A. Either the land has oil, or it does not

Answer 13

given that the land has oil, the test has an 80% chance of saying that the land has oil, do the upper branch will look like so(remember that B is the probability of getting a positive test for oil)
so far so good?

Answer 14

so*

Answer 15

yesir

Answer 16

ok, so the lower branch corresponds to the event that there is no oil, in which case there is a 20% chance the test will say there is oil anyway. this gives us

Answer 17

.8*.55?

Answer 18

which event does that correspond to?

Answer 19

80% positive

Answer 20

that is only half of the event you have described. you are describing this path on the treeas you can see, this is the event that there is no oil on the land, and the test correctly indicates that.

Answer 21

.8*.45?

Answer 22

which event does that correspond to?

Answer 23

a and b

Answer 24

right, so where does that belong in our formula?

Answer 25

numerator

Answer 26

correct
now for the denominator

Answer 27

.2*.45

Answer 28

we will again need the total probability theorem

Answer 29

what event is that?

Answer 30

my bad, .2*.55

Answer 31

ok, what does that represent though?

Answer 32

the fraction denominater

Answer 33

it's not the whole denominator, no
that is only the event in which not A and B, which means you are trying to do
we need the probability of only B on the bottom

Answer 34

which one is that??

Answer 35

\[P(A|B)=\frac{P(A\cap B)}{P(B)}\]how many ways can B occur?