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This question already has an answer here:

Deserted Island

You are stranded on a deserted island with little to no food sources. By your guesses, you will only be able to survive another 10 days. Your only real chance of survival is getting rescued by a boat.

If the probability of seeing a boat in 30 days is 95%, what are the chances of seeing a boat in the next 10 days? Stated differently, how likely are you to survive the next 10 days.

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marked as duplicate by Set Big O, mdc32, JLee, Rob Watts, A.D. May 11 '15 at 18:23

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Let $P(X)$ be the probablity of not seeing a boat in X days.

$P(30) = 0.05$
$P(30) = P(10) * P(10) * P(10)$
$P(10) = \sqrt[3]{0.05}$ = ~0.37
So the probability of seeing it is ~0.63 or 63%

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Short answer:

63.16% Chance of seeing a boat

Reasoning:

You have to consider the opposite chance (of not seeing a boat) on a given day, call it N
100% - (N)^30 = 95%
Solving for N gives us N = 0.904966...
Then plug N into N^10 to give the chance of notseeing a boat in 10 days (~36.84%)
Finally: subtract that from 100% to get ~63.16%

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  • $\begingroup$ Thanks Spencer Kerr - this is correct. Can you explain why you cannot take 95% and divide it by 3. (this is the route most people take, arguing that the probability of 'seeing a boat in 30 days' is equal to 'seeing a boat in the first 10 days' OR 'seeing a boat in between 10-20 days' OR 'seeing a boat in the last 10 days'.) $\endgroup$ – Mark H May 11 '15 at 14:27
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    $\begingroup$ @MarkH This (and the accepted answer) assumes that passing boats follow a Poisson distribution, which isn't specified in the question. It could just as well be the case that the only boat that passes the island does so every 30 days, but one trip in twenty is abandoned due to inclement weather. $\endgroup$ – Joe Lee-Moyet May 11 '15 at 17:15
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If the probability of seeing a boat in 30 days is 95%, then the probability of not seeing a boat in 30 days is 5%. To find out the probability of seeing a boat in any one day, we figure out the 30th root of 0.05, which is about 0.905. This means that there is about a 9.5% chance of seeing a boat on any given day and a reciprocal 90.5% chance of not seeing a boat on any given day.

Solution

The chance of not seeing a boat is about 36.84%, so the chance of seeing a boat and, therefore, being rescued, is 63.16%.

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100%. You state that I can survive 10 days regardless of a boat.

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    $\begingroup$ I understand you're likely joking, but you are only answering the rewording of the question, not the question itself. $\endgroup$ – Spencerkatty May 11 '15 at 15:51
  • $\begingroup$ @SpencerKerr Then the re-wording is incorrect. Consider revising the question since it apparently has two different goals. $\endgroup$ – Adam Davis May 11 '15 at 18:27
  • $\begingroup$ I wasn't actually joking; I saw the question to be one of those where you have a very simple answer obfuscated by a bunch of other irrelevant data. The rewording had nothing to do with it; the 2nd sentence told me all I needed to know. $\endgroup$ – dlchambers May 12 '15 at 16:10
  • $\begingroup$ Also: sorry to all for not putting my answer in the hidden short answer / reasoning boxes. $\endgroup$ – dlchambers May 12 '15 at 16:11
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95%. You can't see a boat from the island between 10 and 30 days from now since you'll be dead, therefore the cited probability must equal the probability of seeing a boat between 0 and 10 days.

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