Oscar has lost his dog in either forest A (with probability 0.4) or in forest B (with probability 0.6).
If the dog is in forest A and Oscar spends a day searching for it in forest A, the conditional probability that he will find the dog that day is 0.25. Similarly, if the dog is in forest B and Oscar spends a day looking for it there, he will find the dog that day with probability 0.15.
The dog cannot go from one forest to the other. Oscar can search only in the daytime, and he can travel from one forest to the other only overnight.
The dog is alive during day 0, when Oscar loses it, and during day 1, when Oscar starts searching. It is alive during day 2 with probability 2/3. In general, for n≥1, if the dog is alive during day n−1, then the probability it is alive during day n is 2/(n+1). The dog can only die overnight. Oscar stops searching as soon as he finds his dog, either alive or dead.
a) In which forest should Oscar look on the first day of the search to maximize the probability he finds his dog that day?
- unanswered
b) Oscar looked in forest A on the first day but didn't find his dog. What is the probability that the dog is in forest A?
- unanswered
c) Oscar flips a fair coin to determine where to look on the first day and finds the dog on the first day. What is the probability that he looked in forest A?
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d) Oscar decides to look in forest A for the first two days. What is the probability that he finds his dog alive for the first time on the second day?
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e) Oscar decides to look in forest A for the first two days. Given that he did not find his dog on the first day, find the probability that he does not find his dog dead on the second day.
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f) Oscar finally finds his dog on the fourth day of the search. He looked in forest A for the first 3 days and in forest B on the fourth day. Given this information, what is the probability that he found his dog alive?
a) Forest A
b) 1/3
c)0.5263
d)1/20
e)0.9722
f)0.1333
can you show how you came up with the answers
To solve these probability questions, we can use the concept of conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred. In this case, we will calculate the probabilities based on the information provided.
a) To maximize the probability of finding his dog on the first day, Oscar should choose the forest with the higher conditional probability of finding the dog that day. In forest A, the conditional probability of finding the dog on the first day is 0.25, while in forest B it is 0.15. Therefore, Oscar should look in forest A on the first day.
b) After searching in forest A on the first day and not finding the dog, we need to calculate the probability that the dog is still in forest A. This is a conditional probability problem. Using Bayes' Theorem, we can calculate the probability:
P(A|Not found) = P(Not found|A) * P(A) / (P(Not found|A) * P(A) + P(Not found|B) * P(B))
P(Not found|A) = 1 - 0.25 = 0.75 (probability of not finding the dog in forest A)
P(A) = 0.4 (probability of the dog being in forest A)
P(Not found|B) = 1 - 0.15 = 0.85 (probability of not finding the dog in forest B)
P(B) = 0.6 (probability of the dog being in forest B)
Substituting the values into the formula:
P(A|Not found) = 0.75 * 0.4 / (0.75 * 0.4 + 0.85 * 0.6)
P(A|Not found) = 0.24 / (0.24 + 0.51)
P(A|Not found) ≈ 0.32
Therefore, the probability that the dog is in forest A after not finding it there on the first day is approximately 0.32.
c) If Oscar flips a fair coin to determine where to look on the first day and finds the dog on the first day, we need to calculate the probability that he looked in forest A. Again, this is a conditional probability problem. Using Bayes' Theorem:
P(A|Found) = P(Found|A) * P(A) / (P(Found|A) * P(A) + P(Found|B) * P(B))
P(Found|A) = 0.25 (probability of finding the dog in forest A)
P(Found|B) = 0.15 (probability of finding the dog in forest B)
Substituting the values into the formula:
P(A|Found) = 0.25 * 0.4 / (0.25 * 0.4 + 0.15 * 0.6)
P(A|Found) = 0.1 / (0.1 + 0.09)
P(A|Found) ≈ 0.526
Therefore, the probability that Oscar looked in forest A given that he found the dog on the first day is approximately 0.526.
d) If Oscar looks in forest A for the first two days and wants to find the probability of finding his dog alive for the first time on the second day, we need to multiply the conditional probabilities.
Probability of finding the dog alive on the second day in forest A = P(Alive on Day 2 in A | Alive on Day 1 in A)
= P(Alive on Day 1 in A) * P(Alive on Day 2 in A | Alive on Day 1 in A)
= (2/3) * (2/2+1) = (2/3) * (2/3) = 4/9 = 0.444
Therefore, the probability that Oscar finds his dog alive for the first time on the second day is approximately 0.444.
e) Given that Oscar did not find his dog on the first day in forest A, we need to calculate the probability of not finding his dog dead on the second day. Since the dog can only die overnight and Oscar did not find the dog on the first day, the dog must still be alive going into the second day.
Therefore, the probability of not finding the dog dead on the second day is 1.
f) If Oscar found his dog on the fourth day of the search, and he looked in forest A for the first 3 days and in forest B on the fourth day, we need to calculate the probability that he found his dog alive.
Probability of finding the dog alive = P(Alive on Day 4 in B | Alive on Day 3 in A)
= P(Alive on Day 3 in B | Alive on Day 2 in A) * P(Alive on Day 2 in B | Alive on Day 1 in A) * P(Alive on Day 1 in B)
= (2/4) * (2/3) * (2/2) = 1/3
Therefore, the probability that Oscar found his dog alive on the fourth day is 1/3.