Derivation of Cryonics Probabilities
Brook Norton, 24 Apr 2020
Introduction:
These derivations were originally completed on 11/27/93. An introduction to the method was published in the Alcor magazine shortly thereafter. On 4/24/20 an independent review of the derivations was completed and determined to be correct per the below message from the reviewer:
This derivation has been verified to be mathematically correct by Zihui Liu, who holds a Bachelor degree from Duke University in Math and Mechanical Engineering. As of 1/2020 he is pursuing his Masters Degree in Aero/Astronautics Department at Stanford University.
The derivations follow.
Sections 1 - 6 present the background principles and nomenclature used in following Sections.
Each of Sections 7 - 13 correspond to an output data column in the spreadsheet. These Sections calculate the odds of events occurring, GIVEN THAT YOU HAVE ALREADY DEANIMATED. They do not including the possibility that you may be cured of old age prior to deanimation. It has been arranged this way because I believe most people think in these terms when considering cryonics. They say "What are my odds of being reanimated if I'm frozen in liquid nitrogen?" Sections 7 - 13 address that question.
Section 14 looks at the bigger picture, by calculating the odds that you will defeat Death by either being cured of old age before deanimation or by being reanimated. Section 14 addresses the question "What are my odds of achieving indefinite lifespan by any means?"
NOTE REGARDING INDEPENDENT DISASTERS: The below derivations use fire, earthquake, and tornado as examples of disasters that would destroy the suspended patient. The mathematics is based on these disasters being independent of each other. So, for example, say you are considering the odds of a fire starting. DO think “What are the odds of a fire starting, not including when started by an earthquake or tornado? What are the odds of a fire starting by vandal, lightning, or other non-earthquake/non-tornado causes?” Do NOT think “The odds of fire are higher because earthquakes and tornados may start fires.”
All probability calculations used in this analysis are based on 2 fundamental rules given in Sections 1 & 2.
------------------------------------------------------------------------------------------------------------------
Section 1:
Rule 1) This rule describes events that are mutually exclusive. That is, if one event occurs, no other event can occur. The sum of the probabilities of each event is 1.0.
Example: Suppose you roll a die. The probabilities regarding which number is rolled are:
#1(1/6) + #2(1/6) + #3(1/6) + #4(1/6) + #5(1/6) + #6(1/6) = 1.0.
Note: “#Num(1/n)" means the probability of the die showing Num is 1/n.
------------------------------------------------------------------------------------------------------------------
Section 2:
Rule 2) This rule describes events that are independent of each other. That is, if one event occurs, it does not affect the odds of another event occurring. The probability that ALL OF THE EVENTS WILL OCCUR is the product of the probabilities that each event will occur.
Example: Consider 3 independently caused disasters. Say that within a 20-yr period, the odds of a fire are 1/20, an earthquake 1/100, and a tornado 1/25. The odds that ALL 3 will occur are: (1/20)*(1/100)*(1/25) = .00002 or 1/50,000.
------------------------------------------------------------------------------------------------------------------
Section 3:
By combining rules 1) & 2) you can determine the odds that: Given several independent events, what are the odds that NONE OF THE EVENTS WILL OCCUR ?
Say that A, B, and C are the odds that each event will occur. In example 1), A = 1/20, B = 1/100, C = 1/25. For each event, there are 2 mutually exclusive outcomes; either the event occurs or it does not occur. (odds that event occurs) + (odds that event does not occur) = 1.0.
Now say that A', B', and C' are the odds that each event will NOT occur. A + A' = 1.0, and similarly for B and C. It follows that A' = 1 - A. In example 1), A(1/20) + A' = 1.0, and A' = 1 - A(1/20) = 1 - .05 = .95.
Now, using rule 2), we can say: the odds that NONE of the events will occur is the product of the probabilities that each event will not occur. Odds that NONE of the events will occur = (A')*(B')*(C') = (1-A)*(1-B)*(1-C). In example 1), odds that no occurrence of fire, earthquake or tornado will occur is (1-1/20)*(1-1/100)*(1-1/25) = (.95)*(.99)*(.96) = .903.
------------------------------------------------------------------------------------------------------------------
Section 4:
It is of interest to know what the odds are that ANY EVENT WILL OCCUR. From rule 1), (odds of any event occurring) + (odds of no event occurring) = 1.0.
Now, from above statement, and using result of Section 3, it follows that:
Odds of ANY (independent) event occurring = 1-(1-A)*(1-B)*(1-C). In example 1), the odds that at least 1 of the 3 disasters will occur in a 20-yr period = 1-(.95)*(.99)*(.96)=1-.903=.097.
------------------------------------------------------------------------------------------------------------------
Section 5: Single time frame and marching time frames:
If events occurring in 1 time period (or time frame) are independent of events occurring in other time frames, Section 4 can be used to determine the odds of an event occurring at any time during several time frames.
The odds that an event will occur at least once over the course of several time frames = 1 - (1 - odds of occurring in time frame #1)*(1 - odds of occurring in time frame #2)*(1 - odds of occurring in time frame #3)*(etc.). In example 1), the odds that a fire will occur at least once in sixty years = 1-(1-1/20, for first 20-yr period)*(1-1/20, for second 20-yr period)*(1-1/20, for third 20-yr period) = 1-(.95)*(.95)*(.95)=1-.857=.103.
------------------------------------------------------------------------------------------------------------------
Section 6: Glossary of terms:
The analysis is broken into 20-year segments; t=0 is when the patient deanimates. All the terms used in the analysis are then evaluated at t=20, 40, 60, etc. Whichever time step is being evaluated is referred to as “now”. For example, at t=60 years, “now” is 60 years after deanimation. “in the last 20 years” would refer to the timespan from 40 to 60 years. “The future” would mean at years greater than 60. After updating all terms at 60 years, the analysis would then march to t=80 years and “now” becomes t=80 years, etc.
For convenience, the abbreviations in this section will be used in following derivations and discussions.
dead.bynow - odds that patient will be destroyed by this time by encountering one of the fatal disasters.
dis.1 to dis.5 - user input odds for each of 5 independent disasters. Odds of occurrence in last 20-yrs.
dis.20 - odds that any disaster occurred in the last 20-yr time frame.
dis.bynow - odds that any disaster occurred at any time prior to now.
frozen - odds that patient will remain frozen till this time.
no.dis.20 - odds that no disasters occur in last time frame.
no.tech.20 - odds that reanimation technology not developed in last time frame.
rean.20 - odds that patient will be reanimated in last 20-yr time frame.
rean.bynow - odds that patient will be reanimated by this time.
tech.20 - affordable reanimation technology developed in last 20-yr time frame.
tech.bynow - user input for: affordable reanimation technology available by this time.
tech&dis.20 - reanimation technology and disaster occur in last 20-yr time frame, either first.
tech.after.dis.20 - reanimation technology and disaster occur in last time frame, disaster first
tech.first.20 - Reanimation technology is developed in last time frame prior to any disaster occurring in last time frame. Technology does occur. Disaster may or may not have occurred.
tech.no.dis.20 - reanimation technology developed in last 20-yr time frame and no disaster occurs in that last time frame.
tech.prior.dis.20 - reanimation technology and disaster occur in last 20-yr time frame, technology first
------------------------------------------------------------------------------------------------------------------
Section 7: dis.20 - odds that any disaster occurred in the last 20-yr time frame.
The user inputs the odds for each of 5 independent disasters. The user specifies the odds that each disaster will occur in the last 20-yr time period.
dis.20 = 1-(1-dis.1)*(1-dis.2)*(1-dis.3)*(1-dis.4)*(1-dis.5)
------------------------------------------------------------------------------------------------------------------
Section 8: dis.bynow - odds that any disaster occurred at any time prior to now.
At time = 0: You may be destroyed very shortly after deanimation due to such disasters as autopsy, deanimation in remote location, suspension procedure is inadequate to preserve necessary brain information, etc. The odds of this occurring are a user input.
For time frames after time=0: (from Section 4) dis.bynow = 1-(1- odds that any disaster occurred prior to last time frame)*(1-odds that any disaster occurred in last time frame)
= 1-(1-dis.bynow, at start of last time frame)*(1-dis.20).
------------------------------------------------------------------------------------------------------------------
Section 9: frozen - odds that patient will remain frozen till this time.
The patient will remain frozen if reanimation technology does not exist yet and he has not been destroyed.
(from Section 3) frozen = (1-tech.bynow)*(1-dis.bynow)
------------------------------------------------------------------------------------------------------------------
Section 10: tech.20 - affordable reanimation technology developed in last 20-yr time frame.
The development of reanimation technology can be described in terms of mutually exclusive events (see Section 1). (odds that tech. was developed prior to last time frame) + (odds that tech. was developed in last time frame) + (odds that tech. will be developed after last time frame) + (odds that tech. will never be developed) = 1.0. Changing nomenclature:
(tech.bynow, at start of last time frame) + (tech.20) + (tech. in future) + (tech. never) = 1.0
Restated another way, with first 2 terms combined:
(tech.bynow at end of last time frame) + (tech. in future) + (tech. never) = 1.0
Comparing above 2 equations we see:
(tech.bynow, at start of last time frame) + (tech.20) = (tech.bynow at end of last time frame)
or
tech.20 = (tech.bynow at end of last time frame) - (tech.bynow at start of last time frame)
For example, suppose the user input the odds of 20% (or .20) for reanimation technology occurring by the start of the last 20-yr period, and 30% chance of occurring by end of the 20-yr period. The odds that reanimation technology was developed in last 20-yr period (tech.20) = .30 - .20 = .10. You could say "The odds that the technology was developed by the end of the last time frame are 30%. However, there is only a 10% chance that the technology was developed in the last 20-yrs, because there is a 20% chance that the technology was developed prior to the last time frame.
One must be careful to interpret this term (tech.20) correctly. "Tech.20" is derived from the user input "tech.bynow" which is structured so that the user estimates each time frame without any knowledge as to whether the reanimation technology was or was not achieved in previous time frames. "Tech.bynow" IS an estimate of whether reanimation technology has been developed at any point between time = 0 and the end of the last time frame. "Tech.20" IS an estimate of whether reanimation technology was developed in the last 20-yr time frame. "Tech.bynow" and "Tech.20" ARE NOT an estimate of whether reanimation technology occurred in the last time frame, given that the technology had not yet occurred at the start of the time frame.
An equivalent probability analysis could be constructed where the user did input odds of reanimation technology being developed in last 20 yrs, given that it had not been previously developed. However, this analysis is not constructed that way.
------------------------------------------------------------------------------------------------------------------
Section 11: rean.20 (not to be confused with "tech.20") - odds that patient will be reanimated in last 20-yr time frame.
To be reanimated in last 20 years, the following 2 conditions must occur:
1) No disasters can have occurred up to the beginning of the last time frame.
2) Reanimation technology is developed in last time frame prior to any disaster occurring in last time frame.
From Section 2: rean.20 = (1-dis.bynow, at start of frame)*(tech.first.20); call this equation A, or eqA.
(Note that it was tempting to construct this argument where condition 1 above is: "patient is still frozen at beginning of the last time frame." However, being frozen is not independent of reanimation technology, condition 2, and so a more complex approach would have to be used than that of EqA.)
From Section 1: Technology and disasters occurring in last time frame can be expressed in 3 ways (eqB, eqC, and eqD below) in terms of mutually exclusive events.
(tech.first.20)+(no technology developed in last frame)+(tech. and disaster occur in last frame, disaster first)=1.0
abbreviating: (tech.first.20)+(no.tech.20)+(tech.after.dis.20)=1.0 eqB
(technology and disaster occur in last frame, technology first)+(technology developed and no disaster occurs in last frame)+(no tech. developed in last frame)+(tech. and disaster occur in last frame, disaster first)=1.0
abbreviating: (tech.prior.dis.20)+(tech.no.dis.20)+(no.tech.20)+(tech.after.dis.20)=1.0 eqC
(no.tech.20)+(technology and disaster occur in last frame, either first)+(tech.no.dis.20)=1.0
abbreviating: (no.tech.20)+(tech&dis)+(tech.no.dis.20)=1.0 eqD
Comparing eqB & eqC: tech.first.20 = (tech.prior.dis.20)+(tech.no.dis.20) eqE
Substituting eqE into eqA: rean.20 = (1-dis.bynow, at start of frame)*[(tech.prior.dis.20)+(tech.no.dis.20)] eqF
Comparing eqC & eqD: tech&dis.20 = (tech.prior.dis.20)+(tech.after.dis.20) eqG
In eqG, it is given that reanimation technology and a disaster will both occur in the last time frame. When they both occur, the odds are 50/50 as to whether technology or disaster occurs first. Therefore:
tech.after.dis.20 = tech.prior.dis.20 eqH
Substituting eqH into eqG: tech&dis.20 = (2)*(tech.prior.dis.20)
Rearranging: tech.prior.dis.20 = (.5)*(tech&dis.20) eqI
Substituting eqI into eqF: rean.20 = (1-dis.bynow, at start of frame)*[(tech.no.dis.20)+(.5)*(tech&dis.20)] eqJ
From Section 2: tech.no.dis.20 = (tech.20)*(no.dis.20) eqK
From Section 1: (no.dis.20)+(dis.20)= 1.0 or no.dis.20 = 1-(dis.20) eqL
Substituting eqL into eqK: tech.no.dis.20 = (tech.20)*[1-(dis.20)] eqM
From Section 2: tech&dis.20 = (tech.20)*(dis.20) eqN
Substituting eqM and eqN into eqJ:
rean.20 = (1-dis.bynow, at start of frame)*{(tech.20)*[1-(dis.20)]+(.5)*(tech.20)*(dis.20)}
= (1-dis.bynow, at start of frame)*[(tech.20)-(tech.20)*(dis.20)+(.5)*(tech.20)*(dis.20)]
= (1-dis.bynow, at start of frame)*(tech.20)*[1-(.5)*(dis.20)] eqO
------------------------------------------------------------------------------------------------------------------
Section 12: rean.bynow (not to be confused with "tech.bynow") - odds that patient will be reanimated by this time.
The odds of reanimation can be described in terms of mutually exclusive events (see Section 1). (odds that reanimation occurred prior to last time frame) + (odds that reanimation occurred in last time frame) + (odds that reanimation will occur after last time frame) + (odds that reanimation will never occur) = 1.0.
Changing nomenclature:
(rean.bynow, at start of last time frame) + (rean.20) + (rean. in future) + (rean. never) = 1.0
Restated another way, with first 2 terms combined:
(rean.bynow at end of last time frame) + (rean. in future) + (rean. never) = 1.0
Comparing above 2 equations we see:
(rean.bynow at end of last time frame) = (rean.bynow, at start of last time frame) + (rean.20)
After many time frames, "rean.bynow" approaches a constant value. This constant value is the odds that reanimation will occur.
------------------------------------------------------------------------------------------------------------------
Section 13: dead.bynow - odds that patient will be destroyed by this time.
The destruction of the patient can be expressed in mutually exclusive terms (see Section 1):
At any given time, the patient must exist in 1 of 3 states: frozen, reanimated, or destroyed.
(remains frozen)+(has been reanimated)+(has been destroyed)=1.0
Abbreviating: (frozen)+(rean.bynow)+(dead.bynow)=1.0
dead.bynow = 1-(frozen)-(rean.bynow)
------------------------------------------------------------------------------------------------------------------
Section 14: live - Overall odds that you will overcome old age and live indefinitely.
After many centuries, you can exist in only 1 of 3 states. (As time goes to infinity, odds of remaining frozen approach 0) (It is assumed for simplicity in this analysis that you aren't killed in an accident at any time after you are cured of old age or reanimated.) The 3 states are mutually exclusive (see Section 1):
(cured of old age prior to deanimation)+(reanimated)+(destroyed)=1.0
Abbreviating: (cured)+(reanimated)+(destroyed)=1.0 As time approaches infinity.
Rearranging: (destroyed)=1-(cured)-(reanimated) Equation A or eqA
(cured) is a user input.
(reanimated) is the odds that you are not cured AND that you are reanimated. From Section 2:
(reanimated)=(1-cured)*(rean.bynow in last time frame) eqB
The subtle difference between (reanimated) and (rean.bynow) should be noted. (rean.bynow) is the odds of being reanimated, GIVEN that you deanimated. (reanimated) is looking at the bigger picture, at the odds you will be reanimated, taking into account you may never deanimate (since you might be cured of old age prior to deanimation).
Sub eqB into eqA: (destroyed)=1-(cured)-(1-cured)*(rean.bynow in last time frame) eqC
It is also true that as time approaches infinity:
(live)+(destroyed)=1.0 or
(live)=1-(destroyed) eqD
Sub eqC into eqD: (live)=1-1+(cured)+(1-cured)*(rean.bynow in last time frame) or
(live)=(cured)+(1-cured)*(rean.bynow in last time frame)