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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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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'.) $\endgroup$
    – Walter
    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. $\endgroup$ 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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