status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not read | reprioritisations | ||
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last reprioritisation on | reading queue position [%] | |||
started reading on | finished reading on |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
Adverb | ||
vorzugsweise | preferably , mainly , chiefly | |
möglichst | preferably | |
besser | better , preferably | |
lieber | rather , better , preferably , sooner |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not read | reprioritisations | ||
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started reading on | finished reading on |
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started reading on | finished reading on |
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started reading on | finished reading on |
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started reading on | finished reading on |
status | not read | reprioritisations | ||
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last reprioritisation on | reading queue position [%] | |||
started reading on | finished reading on |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not read | reprioritisations | ||
---|---|---|---|---|
last reprioritisation on | reading queue position [%] | |||
started reading on | finished reading on |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
Harmonic Mean
The harmonic mean of n numbers x_{i} (where i = 1, 2, ..., n) is:
The special cases of n = 2 and n = 3 are given by:
and so on.
For n = 2, the harmonic mean is related to arithmetic mean A and geometric mean G by:
status | not read | reprioritisations | ||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not read | reprioritisations | ||
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started reading on | finished reading on |
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started reading on | finished reading on |
status | not read | reprioritisations | ||
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last reprioritisation on | reading queue position [%] | |||
started reading on | finished reading on |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not read | reprioritisations | ||
---|---|---|---|---|
last reprioritisation on | reading queue position [%] | |||
started reading on | finished reading on |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|
repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
The two defining properties of a probability are as follows:
The probability of any event E is [...]
The sum of the probabilities of any set of mutually exclusive and exhaustive events equals 1.
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
The two defining properties of a probability are as follows:
The probability of any event E is a number between 0 and 1: 0 ≤ P(E) ≤ 1.
The [...]
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
Covering or containing all possible outcomes.
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
P followed by parentheses stands for [...]
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
If A, B, and C are mutually exclusive and exhaustive events then P(A) + P(B) + P(C) = [...]
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
The probability of any event is the [...] included in the definition of the event.
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
In investments, we often estimate the probability of an event as a [...]
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
The probability of an event estimated as a relative frequency of occurrence is called an [...]
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
Relationships must be stable through time for [...] to be accurate.
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scheduled repetition interval | last repetition or drill |
Can we ecalculate an empirical probability of an event not in the historical record?
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Can we calculate a reliable empirical probability for a very rare event?
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
A probability drawing on personal judgment is called a [...]
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
There are cases in which we may adjust an empirical probability to account for perceptions of changing relationships in this cased we are talking about a [...]
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
There are cases in which we have no empirical probability to use at all, then we would be making a [...]
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Subjective probabilities appear in various places in this reading, notably in our discussion of [...]
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
Because a priori and empirical probabilities generally do not vary from person to person, they are often grouped as [...]
status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
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In business and elsewhere, we often encounter probabilities stated in terms of odds—for instance, “the odds for E” or the “odds against E.” For example, as of November 2013, analysts’ fiscal year 2014 EPS forecasts for JetBlue Airways (NASDAQ: JBLU) ranged from $0.55 to $0.69. Suppose one analyst asserts that the odds for the company beating the highest estimate, $0.69, are 1 to 7. Suppose a second analyst argues that the odds against that happening are 15 to 1. What do those statements imply about the probability of the company’s EPS beating the highest estimate? We interpret probabilities stated in terms of odds as follows:
Probability Stated as Odds. Given a probability P(E),
Odds for E = P(E)/[1 − P(E)]. The odds for E are the probability of E divided by 1 minus the probability of E. Given odds for E of “a to b,” the implied probability of E is a/(a + b).
In the example, the statement that the odds for the company’s EPS for FY2014 beating $0.69 are 1 to 7 means that the speaker believes the probability of the event is 1/(1 + 7) = 1/8 = 0.125.
Odds against E = [1 − P(E)]/P(E), the reciprocal of odds for E. Given odds against E of “a to b,” the implied probability of E is b/(a + b).
The statement that the odds against the company’s EPS for FY2014 beating $0.69 are 15 to 1 is consistent with a belief that the probability of the event is 1/(1 + 15) = 1/16 = 0.0625.
To further explain odds for an event, if P(E) = 1/8, the odds for E are (1/8)/(7/8) = (1/8)(8/7) = 1/7, or “1 to 7.” For each occurrence of E, we expect seven cases of non-occurrence; out of eight cases in total, therefore, we expect E to happen once, and the probability of E is 1/8. In wagering, it is common to speak in terms of the odds against something, as in Statement 2. For odds of “15 to 1” against E (an implied probability of E of 1/16), a $1 wager on E, if successful, returns $15 in profits plus the $1 staked in the wager. We can calculate the bet’s anticipated profit as follows:
Win: | Probability = 1/16; Profit =$15 |
Loss: | Probability = 15/16; Profit =–$1 |
Anticipated profit = (1/16)($15) + (15/16)(–$1) = $0 |
Weighting each of the wager’s two outcomes by the respective probability of the outcome, if the odds (probabilities) are accurate, the anticipated profit of the bet is $0.
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
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scheduled repetition interval | last repetition or drill |
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||
scheduled repetition interval | last repetition or drill |
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