Probabilistic inference vs statistical inference

[curious_mind]

I wanted to understand when do we need probability and statistics together and when either probability or statistics can be used independently to derive inference

 

[wwfeldman]

i will make an attempt at your question
i am not an expert in either probability or statistics

statistics are derived from data
an example is test scores
your class takes a test, from that, a variety of statistics may be produced:

maximum score
minimum score
mode (most common score)
mean (or average) score
median (the middle score found by eliminating the highest and lowest and repeating until one (or two) scores are left
standard deviation (+/- range for about 67% of the grades)

there are a lot of other statistics available that measure other qualities of the data

probability is derived from speculation
an example is picking a card from a standard deck of playing cards (solitaire, bridge or poker, not euchre)

probability is measured on a scale of 0 to 1

you pick one card from the deck
what is the probability that:
it will be a club? 13 out of 52 or 1/4
it will be a 9? 4 out of 52 or 1/13
it will be a 9 of clubs? 1 out of 52 or 1/52
it will be a card? 52 out of 52 or 1

as to your question, we can clearly use probability and statistics separately

they inform each other
you can find the results of picking cards by inferring from the nature of the deck of cards
or you can do the experiment and get statistics

some things, like cards, are relatively easy to calculate probabilities
some things, require looking at the available data, like test scores

 

[rayweathers79]

Well, let's put it in gambling terms.

All events in gambling games have absolute probabilities that depend on sample spaces or the total number of possible outcomes. For example, if you toss a six-sided die, the sample space is six, with the probability of landing on any particular side one in six. Games with huge sample spaces, like poker, have events with small probabilities. For instance, in five-card poker, the probability of drawing four of a kind is 0.000240, while the chance of drawing a royal flush, the rarest hand, is a mere 0.00000154.

Skilled poker players understand the sample spaces of the game and probabilities associated with each hand. Thus, estimating the odds of a particular hand will guide their gamble.

Adept players are interested not only in probabilities but also in how much money they can theoretically win from a game or event. The average amount you can expect to win is aptly called the expected value (EV), and it is mathematically defined as the sum of all possible probabilities multiplied by their associated gains or losses.

Generally, skilled gamblers assess the risk of each round based on the mathematical properties of probability, odds of winning, expected value, volatility index, length of play, and size of the bet. These factors paint a numerical picture of risk and tell the player whether a bet is worth pursuing.

Still, gambling involves far more than simple mathematical properties. Gamblers use a great deal of social psychology to read their fellow players. The ability to decipher bodily cues, for instance, helps discern fellow players’ mental states and possibly gives a clue to the statistics of their hands.

Gambling is an art and a science; only the best players can synthesize the two to reap millions.