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# Probability & Statistics/Science fiction story with maths odds

Question
Hi,

I recently read a science-fiction story about a man, in order to avoid the usual unemployment in society, goes in for a game with an organ-transplant company wherein he has to visit a special room once every week and drink from just one bottle out of a selection of fifty bottles in that very room. 49 of those bottles just contain a standard sleeping draught which makes him at first drowsy and then sleep for a few hours - afterwards he collects 1000 US Dollars for that experiment, and then he can try the same game the next week for the same prize. One of those 50 bottles, however, also makes him, at first drowsy, and then asleep, but the "sleep" is then permanent -all bottles taste the same and provide the same effect/sensations on the subject before loss of consciousness. If he drinks the wrong bottle, his body is then placed into suspended animation in cryogenic tanks and his organs are then subsequently sold to the highest bidder. My question is, how safe would this practice be in general and what would the odds be of  doing this experiment weekly and living a full life from, say, aged 20 to 100 without becoming fodder for the organ-vats.Thanks!

Well, first of all, this game isn't very safe at all. If you spin a roulette wheel and land on 00, they don't shoot you. There would be thousands of people killed at casinos all over the world. This would be extremely unethical.

I know this is coming from a fictional book, but I should make it clear: this isn't safe, I don't endorse or condone this, and it violates scientific and medical ethics, not to mention the law in virtually every country.

Now, that being said, in a fictional context, how does this work?

Well, every week is the same -- a 1/50 chance of dying. (And once you die, it's over.)

Now, to live two weeks, what are the odds? Well, you have to live twice, so you multiply:

(49/50) × (49/50) ≈ 96%

To live for a full year, you would have to live for 52 trials (or maybe 53) in a row:

(49/50)^52 ≈ 34%

That's nearly 1/3 chance of surviving a single year. To live 80 years, you'd have to live through about 4171 trials. The probability of that is:

(49/50)^4171 ≈ 2 × 10^(-37)

That's roughly one in four million million million million million million. I do not like those odds!

If "p" is the chance of dying, 1-p is the chance to survive. After N weeks, your chance of survival is therefore:

(1-p)^N

Assuming p is not zero, this is going to be exponentially bad . Often the word "exponential" is used incorrectly or figuratively, to exaggerate how good or bad something is. Here, it is literally exponentially bad (because N is in the exponent). Over time, your chance to survive is terrible.

It might sound good for a week, or maybe two weeks, but this isn't sustainable. Virtually no one would survive a lifetime. Let's work that out:

If we conducted a trial and logically expected a single person to five years, how many people would be in that trial? (The answer will be shockingly large.)

The number of people in the trial could be M. If a single person survives with (1-p)^260 probability (260 weeks in 5 years), then the "average" or "expected" number of survivors is:

M × (1-p)^260 = 1

In other words, if we want exactly one person to survive, we start with 191 people. Can you imagine a drug trial, military campaign, or natural disaster involving nearly 200 people and only one survives? Shocking!

In the end, a better question might be this: How many bottles would you have to use to make longevity possible? Let's say you wanted everyone to have a 50/50 chance of surviving 80 years of trials. Then what you really want is:

(1-p)^4171 = 1/2

To solve for this new value of p, you get p ≈ 0.000166169. In order to achieve a death-rate less than that, you'd have to use over 6000 bottles instead of 50.
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks. The SF story I mentioned had sort of falsely assumed, on behalf of the reader or the needs of the story-plot, that this game was a safe long-term bet for one person, to the point where the main character in the story is emotionally-tricked by the company into doing another game, where he has to choose only one from 2 bottles in the room to get a chance at 25,000 US dollars, one of the 2 bottles being the deadly one.Thanks for being so detailed in the answer and reminding me of the maths that I had obviously not so well recalled from school..

Probability & Statistics

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#### Clyde Oliver

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