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!
To calculate the probability in these scenarios, we can use the concept of conditional probability and basic principles of probability.
i) To find the probability that a Hobbit believes in Ents given that they believe in Elves, we use Bayes' Theorem. Bayes' Theorem states that P(A|B) = P(A∩B) / P(B), where P(A|B) represents the probability of event A occurring given that event B has occurred, P(A∩B) represents the probability of both A and B occurring, and P(B) represents the probability of event B occurring.
In this case, event A is believing in Ents, and event B is believing in Elves. From the given information, we know that 9% of Hobbits believe in both Elves and Ents. Therefore, P(A∩B) = 0.09.
Also, we know that 28% of Hobbits believe in Elves, so P(B) = 0.28.
Plugging these values into Bayes' Theorem, we can calculate P(A|B):
P(A|B) = P(A∩B) / P(B) = 0.09 / 0.28 ≈ 0.3214
Therefore, the probability that a Hobbit believes in Ents given that they believe in Elves is approximately 0.3214.
ii) Assuming independence, the probability that both Hobbits believe in Elves can be found by multiplying the individual probabilities. Let's assume the probability of a Hobbit believing in Elves is P(E), which is 0.28.
Using the principle of independence, the probability that both Hobbits believe in Elves is:
P(both believe in Elves) = P(E) * P(E) = P(E)^2.
P(both believe in Elves) = 0.28 * 0.28 = 0.0784
Therefore, the probability that both Hobbits believe in Elves is 0.0784.
Regarding your personal question, probabilities do indeed add up to 100% or 1 when considering all possible outcomes. In this case, the probabilities provided are not exhaustive, as they only consider belief in Elves and Ents. The remaining percentages may represent Hobbits who believe in other things or have no beliefs about these mythical creatures.
Supposing the question asks to find the probability that the Hobbit does not believe in Ents, you are correct that you would subtract the probability of believing in Ents (0.16) from 1. So the probability that the Hobbit does not believe in Ents would be 1 - 0.16 = 0.84, which is 84%.
It's important to note that probabilities should always be interpreted in their appropriate context, and the given percentages may not represent the complete set of probabilities for all beliefs or lack thereof among Hobbits.