Suppose that you visit Shire, and find that 28% of Hobbits believe in the existence of Elves, 16% believe in the existence of Ents, and 9% believe in the existence of both Elves and Ents.
You come across a Hobbit:
i) What is the probability that he will believe in Ents given that he believes in Elves?
ii)Suppose you come across two Hobbits:
What is the probability that they will both believe in Elves? (Assume Independence)
Personal Question: I thought probabilities always add up to 100% or 1, however these add up to 53. Does it not matter? Supposing the question asks, find the probability that the hobbit(assuming you come across one) does not believe in Ents, do you subtract .16 from 1? I just need clarification. Thank you!
conditional probability:
prob (ents given elves)
= prob(endts | elves)
= prob(elves AND ents)/prob (elves)
prob(ents | elves)
= .09/.28 = .3214
prob (2hobbits believing in elves)
= (.28)(.28) = .0794
They do add up to 1 if you add up ALL possible cases, you added up only 3
A good way is to look at the problems with Venn diagrams
Draw 2 intersecting circles , label one elves, the other ents.
put .09 in the intersection of the two.
now look at the "elves" circle, it is supposed to contain .28, but we have already counted .09, so place .19 in the part of the circle not intersecting with the other one.
look at the "ents" circle, it must contain .16 but we already counted .09, leaving .07 for the rest of that circle
Count up the 3 numbers to get .35
That leaves .65 outside of both circles.
So now we have 4 cases
hobbits believing in neither one = .65
hobbits believing in both = .09
hobbits believing in only elves = .09
hobbits believing in only ents = .07
add them up to get 1
When you added .28+.16+.09
wasn't the .09 contained in both .28 and .16 ?
So you added it twice, that's why it made no sense
near the end it should say:
So now we have 4 cases
hobbits believing in neither one = .65
hobbits believing in both = .09
hobbits believing in only elves = .19
hobbits believing in only ents = .07
add them up to get 1
To find the probability of a Hobbit believing in Ents given that they believe in Elves, we can use conditional probability.
i) Let's denote the event of believing in Elves as E and the event of believing in Ents as T. We are given that P(E) = 28% and P(E ∩ T) = 9%.
The probability that a Hobbit believes in Ents given that they believe in Elves is denoted as P(T|E). We can calculate this using the formula:
P(T|E) = P(E ∩ T) / P(E)
Substituting the given values, we get:
P(T|E) = 0.09 / 0.28 ≈ 0.3214 (approximately 32.14%)
So, the probability that a Hobbit will believe in Ents given that they believe in Elves is approximately 32.14%.
ii) For the second question, we assume independence between the two Hobbits. This means that the belief of one Hobbit does not influence the belief of the other.
Assuming the probability of a Hobbit believing in Elves is denoted as P(E), the probability that both Hobbits believe in Elves is given by P(E) * P(E) = P(E)^2.
Since the probability of each Hobbit believing in Elves is 28%, we can substitute this value into the equation:
P(both Hobbits believe in Elves) = 0.28 * 0.28 = 0.0784 (approximately 7.84%)
Therefore, the probability that both Hobbits believe in Elves is approximately 7.84%.
Regarding your personal question:
Probabilities should indeed always add up to 100% or 1. However, in this case, we are dealing with probabilities of two different events (belief in Elves and belief in Ents), so they are not directly related to each other. Probability values can exceed 1 or 100% when considering events independently.
For example, in this scenario, a Hobbit could believe in both Elves and Ents simultaneously, so the probability of believing in both can be more than 1%. It's important to note that these probabilities are not mutually exclusive.
If the question asks to find the probability that a Hobbit does not believe in Ents, then you would subtract the probability of believing in Ents from 1. In this case, P(not T) = 1 - 0.16 = 0.84 (or 84%) would represent the probability that a Hobbit does not believe in Ents.
I hope this clarifies your questions! Let me know if there's anything else I can assist you with.