Terminologies

Experiment: Has a well defined outcome, for example, tossing a coin we get Heads or Tails. Tossing a coin 20 times and recording the 20 outcomes is a single experiment with 20 trials. The probabilities we get after conducting experiments are called experimental probabilities, and are easier to compute than true probabilities.

Sample Space: Set of all possible outcomes of an experiment. For example, sample of a dice role is ${1,2,3,4,5,6}$

Event: Outcome (or combination of outcomes) of a random experiment. For example, getting a Head when we toss a coin is an event.

Mutually Exclusive Events: These are events that CANNOT happen simultaneously. For example, when we roll a dice, we cannot get both heads and tail at the same time, so getting a head or a tail are mutually exclusive events.

Expected Value: The average outcome we expect if we run an experiment many times. $E(A)$ denotes the expected value of an event $A$.

• For categorical outcomes, $E(A) = P(A).n$, where $n$ is the number of trials. As an example, if I draw a card at random from a deck of cards, record the outcome, put it back and repeat this 20 times, the expected value of getting a spade is $E(A) = (13/52)*20 = 5$, meaning we expect to get a spade 5 times if we run this experiment. These is an expectation, not a guarantee.
• For numerical outcomes, we multiply the outcome with its probability, we then add all the values to get the expected value. Example: if we are hitting a target, and the outer layer is worth 10 points, middle later is 20 points and bulls eye is 100 points. Probability of hitting each layer is 0.5 (outermost), 0.4 (middle) and 0.1 (center), therefore $E(A) = (0.510)+(0.420)+(0.1*100) = 23$, however, this isn’t a value we will get when we hit the target board. Expected values are used to make predictions about the future based on past data, and these predictions are made based on intervals instead of specific values due to the uncertainty the future holds.

Simple Event: Events with one outcome, example, rolling a dice.

Definition

The probability of an event E occurring is defined as follows:

$$ \large P(E) = \frac{n(E)}{n(S)}, \; 0 \le P(E) \le 1 $$

$n(E)$ = Number of ways in which event E can happen

$n(S)$ = Number of possible outcomes of the experiment

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Probabilities are only indications of how likely events are; they’re not guarantees.Therefore, when we roll a dice and if we are interested in getting a 6:

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