How likely is it then that we answer at least 5 questions correctly? This probability must be higher since we also can answer 6, 7, 8, 9 or 10 questions correct. Therefore we must add the probability that we guess 6, 7, 8, 9, and 10 questions correct.
In how many ways can we be right on 5 questions? 10!/5!(10-5)! = 252 ways In how many ways can we be right on 6 questions? 10!/6!(10-6)! = 210 ways In how many ways can we be right on 7 questions? 10!/7!(10-7)! = 120 ways In how many ways can we be right on 8 questions? 10!/8!(10-8)! 45 ways In how many ways can we be right on 9 questions? 10!/9!(10-9)! = 10 ways Inhowmanywayscanweberighton lOquestions? 10!/10!(10-10)!= 1 way
Total = 638 ways
Since each guess has a 50% probability of being right and there are 10 questions and we want to be right on at least 5 of them and there are 638 equally likely ways we can answer at least 5 questions, the probability that we answer at least 5 questions right is (0.5)5 x (0.5) 5 x 638 = 62.3%.
The example illustrated a binomial experiment. The probability distribution for a binomial experiment is: Number of possible ways of selecting k things from n things (observing k successes inn trials) x (probability of success)k (1 - probability of success)n-k
If we put in the above figures we get: 252 x (0.5)5 x (0.5)5 + 210 x (0.5)6 x (0.5)4 + 120 x
(0.5)7 x (0.5)3 + 45 X (0.5)' X (0.5)2 + 10 X (0.5)9 X (0.5)' + lx (0.5)' 0 X (0.5)0 = 62.3%
Binomial experiments have the following characteristics: An event that is repeatable or the experiment consists of n number of identical and independent trials. There are only two outcomes in every trial - success/failure, right/wrong, present/absent, 0/1 etc. and the probabilities of success and failure are constant in every trial.
Examples of binomial experiments are firing a projectile at a target (hit/miss), developing a new drug (effective/not effective), dosing a sale (sale/no sale) etc.
We toss a single die 5 times. How likely is it that you will roll exactly 3 sixes? What is success? Rolling a 6 on a single die. What's the probability of rolling a 6 on a single die? 1/6 (there are 6 outcomes and one of them is a success). What is the probability of failure? 1-1/6 = 516. What is the number of trials? 5. What is the number of successes out of those trials? 3. In how many ways can you roll three sixes (successes) in 5 trials? 5!/3!(5-3)! = JO Probability= JO x (116)3 x (5/6J = 3.2%
A boat has three independent engines and needs at least two to work properly. The probability that each engine works properly is 98%. The probability that all three engines work is 94.1%
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) .
(0.983 The probability that at least one engine fails (either engine 1, 2 or 3) is therefore 5.9%
(this is the same as the probability that exactly 1 engine fail + exactly 2 engines fails + exactly 3 engines fails).
What is the probability that at least 2 engines work? Let's go back to combinations and the binomial distribution: probability (3 engines work) + probability (2 engines work) = 3!/3!(3-3)! x (0.98)3 x (0.02)" + 3!/2!(3-2)! x (0.98J x (0.02)1 = 99.8816%. The probability that at least two engines will fail is therefore 0.1184%. The boat will fail to work in 1 out of 845 times.
Let's add a backup engine. What is now the probability that at least 2 engines work? Probability (4 engines work) + probability (3 engines work) + probability (2 engines work) = 4!/4!(4-4)! x (0.98)4 X (0.02)" + 4f/3f(4-3)! X (0.98)3 X (0.02)1 + 4f/2f(4-2)f X (0.98/ X (0.02/ =
99.996848%. The probability that at least three engines will fail is therefore 0.003152%. The boat will now only fail in 1 out o/31,726 times.
Binomial probabilities assume independence. It may be that the failure of one engine increases the failure probability of a second engine. For example, the failure of one engine increases the load on which the second engine is run. Using one engine causes more stress and wear to the second engine, etc.
Calculations to some of the examples
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150 The possible number of ways we can choose 6 numbers out of 49 are 49!/(49-6)!6!
= 13,983,816.
24 hours equals 1,440 minutes. One 365-day year equals 525,600 minutes. 14 million minutes equals about 27 years.
The probability we succeed is (0.8) 6 or 26%.
)
The probability that IO mutually independent start-ups all succeed is 0.01% (0.410
) .
but the probability that at least one succeeds is 99.4% (I - 0.610
) .
The probability that at least one of the parts don't work is 86.5% (1 - 0.999 2000 Assuming independence, the probability of system failure (where at least one part must fail for the system to fail) is 1 minus the reliability of the system.
161 Assuming independence, the probability of system failure (where both navigation systems must fail for the system to fail) is the product of the probabilities of primary and back up system failing.
162 An event that has one chance in 20 of happening in any given year is nearly certain to happen over 50 years (I - 0.95 50 = 92.3%). If there is a 5% chance that an event happens in any given year, then the chance that it won't happen is 95%. The chance that it won't happen over 50 years is 7.7%. This means that the probability that the event happens at least once is 92.3%.
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162 The probability that at least one accident will happen in any given year is 3.9% {1-
0.99940). The probability that at least one accident will happen during the next 10 years is 33% (1 - 0.96110 ).
162 The probability of a major earthquake happening in any given year (assumed to be a constant) is therefore 3.2% ((1-p)30 = 38%). The probability that a major earthquake
) .
will happen at least once during the next 5 years is 15% (1 - 0.9685
165 In a group of 1,048,576 (220) people it happens to someone. In fact, in the U.S. a country with 280 million people, one in a million chance events happen 280 times a day (1/1,000,000 x 280 million).
165 1 person has 365 possible birthdays if we assume that there are 365 days to choose from and that all birthdays are equally likely to happen. When there are 2 persons in a group the second person can choose among 364 possible birthdays that are not shared with the first person. The second person only shares 1 day with the first person. The chance that 2 persons share birthdays is therefore 1 out of365 or 0.27%. When there are 3 people in a group it is easier to find the likelihood that 2 of them share a birthday by first finding out how likely it is that no one of these 3 people share birthdays. When there are 3 people in a group the third person can choose among 363 possible birthdays that are not shared with any of the first 2 persons. This means that the chance that the third person will not share birthday with any of the first 2 persons is 363 out of365 or 99.45%.
To find the probability that multiple events happen, we multiply the individual probabilities together. The probability that no one in a group of 3, share birthday is therefore: 365/365 x 364/365 x 363/365 = 99.18%. Therefore, the probability that 2 people in a group of 3 share birthdays is 1 - 0.9918 or 0.82%. Let's repeat this procedure for a group of 23 people:
365 X 364 X 363 X ••••••343 = 49.30/o.
365 23
Therefore, the probability that 2 people in a group of 23, share birthdays is 1 - 0.493 or 50.7%.
178 At the end of 10 predictions, one monkey has a perfect record of predicting the direction of interest rates (1,000 x 0.510).
180 The number of ways to get 2 successes in 10 trials are 10!/ (10-2)!2! or 45. The probability is 45x (O.8 )2 x (O. 2)8 or 0.007%.
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- APPENDIX FOUR -
CHECKLISTS
Helpful for achieving goals, making choices, solving problems, evaluating what is likely to be true or false etc.
Use notions
Use the big ideas that underlie reality
Understand what something really means
Simplify
Use rules and filters
Know what I want to achieve
Find and evaluate alternatives<
br />
Understand consequences and their consequences on the whole
Quantify
Search for and base things on evidence
Think things through backward
Remember that big effects come from large combinations of factors
Evaluate the consequences ifI'm wrong
What is the issue?
What's the question? What is this really about?
What's the essence or nub of the issue? What is then the key question?
Relevant? Solvable? Important? Knowable? Utility- applicability?
Do I understand what the subject is all about? In order to have an opinion on a subject I need some relevant data and basic knowledge about the subject, otherwise just say: "I don't know".
Is my judgment here better than others?
What must I predict here and is it predictable?
Is a decision needed? What happens if I don't deal with this? Is this something I can do anything about? Should "I" do this?
Over what period of time am I considering this issue? Where am I at present? From whose point of view?
Simplify by deciding big "no-brainer questions" first and begin from where I am.
Understand what it means
Translate words and ideas in terms I understand. Do I understand what words and statements really mean and imply? Does it mean anything? Will it help me make useful predictions on what is likely to happen?
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- Do I understand how and why something works and happens? What is it doing? Why does it do that? What is happening? How and why is this happening? What is the consequence of this (observation, finding, event, experience...)?
- Definitions and implications?
Filters and Rules
- Use filters incl. rules and default rules - what test(s) can I make?
- Adapt to my psychological nature, abilities, advantages, and limitations
- Consider values and preferences and therefore priorities and what I want to avoid
What do I specifically and measurable want to achieve and avoid and when and why?
- What future "value" do I want to achieve? Target numbers? Target effects? Time horizon?
- Assume I have already reached my target. What would this imply in numbers and effects? What must then have been achieved? Is it (target) reasonable? Is it reasonable ifI reverse this to the present?
- Do I have ways to measure to what degree my goal is being achieved? Key variables or components of yardstick?
- IfI achieve this what will happen? Do I want that to happen?
- Can I break my goal into short-term goals with deadlines?
- What is my real reason for doing this? Is it because I want to or because I have to? Have I stated my goal from internal and external realities or am I biased now or influenced by some psychological forces?
- Can I express my goal in a way that makes it easier to see how it is going to be achieved?
- Is this the true goal for what I want to achieve?
What is the cause of that?
- To achieve my goal I must understand what causes my goal to be achieved
- What is the equation for goal and what evidence do I have for that?
- What don't I want to achieve? What causes non-goal and what can I do to avoid that? What must I not do or what must I avoid?
- What variables influence the system? What are the critical forces and variables, the ones that account for the main outcome? What is the key unknown? What is the certainty with which I can evaluate, optimize ... the different variables?
- Which variables are dependent on other variables (or situation, environment, context, timing, behavior) and which ones act independently of each other?
- What force causes a variable to be achieved? What produces the force(s)? Are there short and long-term forces? What is their relative strength? How do they combine and interact and what are the effects? How can I get many forces to operate together in the same direction? What lack of force would destroy the system? What produces this force? How predictable are they? What can the forces in place rationally expect to cause? What forces are temporary and which ones are permanent? How will the system change as the forces that act on the variable change?
- How resistant is the system to a change in the variables and/or forces? What are the likely
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wanted and unwanted short- and long-term consequences (on numbers and effects) of changes (up/down) - scale, size or mass, strength, intensity, length, time horizon, environment, participants etc. - in the variables or forces? What happens when a number of small causes operate over a long time? What are the consequences if a force acts on the variable over a long time? Which force could change it? What is needed to create a critical mass? What forces when added can create a critical mass? How? Will something else happen when I change a variable or force? What must happen for a force to change? Can a change produce other consequences (observe that I am interested in effects on the whole system and the final outcome)? Does a change in one variable make a dramatic difference in outcome? Will the properties also change? What are the consequences if the relation between the variables changes? What is the change point? Barriers? Catalyst? Tipping point? Inflection point? Break point? Limits? Is there a time lag before effects happen? Feedback? What can speed up the cause? What are the critical points when the effects get reversed? What can I change in the equation and what can others change? How? Who? When? Which variables must I change to achieve goal? How can I measure the amount of change? Degree of sensitivity ifl change the assumptions? Effects on goal and path? What will happen ifl hold one variable constant? Ifl at the same time increase one variable and decrease another? Net effect? Ifl change one variable or force at a time? What is it in the environment that can change the situation? What other advantages and disadvantages can I achieve if I optimize one of the variables? What must happen to cause a change in outcome? Is it still a variable ifl change the conditions?
- Are there exceptions to the equation and why? What conditions are required to achieve goal? Has my goal different cause short-term and long-term? Is the cause time-dependent? Can I deduce the cause by observing the effects? Have I looked upon the system from different angles and viewpoints? What does the measurement of the subject depend on?
- What is the major constraint that limits the goal from being achieved?
What available alternatives do I have to achieve my goal?
- Judge alternatives in terms of goal, subject in question, rules and filters, cause and effect, human behavior, evidence, counter evidence, simplicity and opportunity cost of money, time, other resources, effort, understanding, risk and mental stress.
- What evidence (incl. models) do I have that these alternatives are most likely to achieve goal?
- Are they depending on time horizon or event?
- What are the likely consequences of each action? What possible outcomes can happen? Likelihood? How desirable is each consequence?
- Do I forego any future opportunities ifl make a specific action now?
What are the consequences?
- Find out which alternative are most likely to achieve my goal by estimating their likely consequences
- If I do this, what will happen? Why will this not happen?
- What are the likely (logical) wanted and unwanted (or unintended) consequences
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(quantitative and qualitative) and consequences of consequences (immediate and over the course ofa long period) of each alternative/event (proposition) factoring in relevant variables?
- What are the different scenarios and outcomes that can happen? What is likely to happen short and long-term based on the evidence?
- What can help me make predictions on consequences or if something is likely to be true or false?
- What must happen for the goal to be achieved? How likely is it that the necessary events will happen and happen to me? What does the probabilities favor? What will happe
n ifI reverse the proposition?
- What are the uncertainties that can significantly influence the outcome? What unintended consequences are there due to repeating effects, complications ... ? Is the net effect positive? Does the consequences predict anything else? What else does this mean?
- What are the consequences if this is true or false?
- Have I considered the whole system from different viewpoints? Have I considered social, financial, physical and emotional consequences? What are others likely to do? What are my experiences of earlier behavior? What happens when others do the same?
Bias
- Is there any reason for bias due to self-interest or psychological influences that may cause misjudgment?
- Is this a biased statement or a fact? What are factual judgments and what are value judgments?
- How reliable is he? Is he competent enough to judge? Credentials? What is his purpose with this? Does he have any motive to lie? How does he know that this is true?
Seeking Wisdom Page 40