• Probability is a measure of the likelihood of an event to occur.
• Probabilities are expressed as numerical values between 0 and 1, where 0 indicates impossibility (an event will not occur), 1 indicates certainty (an event will occur), and values between 0 and 1 represent degrees of likelihood.

A possible result that could occur during an experiment.

An event is an outcome or a set of outcomes of an experiment or a random process.

The sample space is the set of all possible outcomes of an experiment.

A trial refers to a single occurrence or experiment in which an outcome is observed.

The sample space is ${1, 2, 3, 4, 5, 6}$.

The probability of an event A, denoted as $P(A)$, is the likelihood that event A will occur.

$P(A) = \frac{\text{Number of outcomes for event A}}{\text{Total numer of possible outcomes}}$

1. Experimental: Probability of something happened based off the results of an actual experiment
2. Theoretical: The expected probability of something happening.

As the number of trials increases, the experimental probability will get closer to the theoretical probability value.

• Definition: An event is affected by other events. The probability of the second event is conditional on the outcome of the first.
• Example: The probability of drawing a king on the second draw from a standard deck of 52 cards without replacements. The probability of drawing a king on the second draw depends on the outcome of the first card.

• Definition: Each event does not affect each other. The outcome of the first event has no impact on the probability of the second event.
• Example: Tossing a fair coin twice is an example of independent events. Whether the first coin toss results in heads or tails does not influence the outcome of the second toss.

• Definition: Events that cannot occur simultaneously.
• Example: Not possible to flip heads and tails simultaneously.

• The complement of event A $(A’)$ , is the set of outcomes that are not in A.
• The probability of $A’$ is $1-P(A)$

The intersection of events A and B $(A \cap B)$ is the set out outcomes that are in both A and B.

The union of events A and B $(A \cup B)$ is the set out outcomes that are in either A or B or both.

The gambler’s fallacy is the mistaken belief that if a particular event occurs more frequently than normal recently, it is less likely to happen in the future (or vice versa), given that the probability of such events does not depend on what has happened in the past.

Regression to the mean implies that following an extreme random event, the next random event is likely to be less extreme. However, the extreme event may be still above the mean. In contrast, gamblers fallacy implies that the next random event should be below the mean (expected value) to even out the extreme random event that was above the mean, which is a wrong belief.

Marginal probability (aka simple probability) refers to the probability of a single event occurring, without considering the influence of other events or variables.

Definition: Conditional probability is the probability of an event occurring given that another event has already occurred.

Formula:

$$ P(A|B) = \frac{P(A\cap B)}{P(B)} $$

Where

• $P(A|B)$: Conditional probability of event A given event B.
• $P(A\cap B)$: Intersection Probability.
• $P(B)$: Probability of event B occurring.

$P(A|B) = P(A)$

The conditional probability of A given B is the marginal probability of A because event B does not influence the probability of event A.

$P(A \cap B) = P(A)\times P(B)$

Multiply the individual probabilities to find the probability of both events occurring simultaneously.

$P(A \cap B) = P(A) \times P(B|A)$

Multiply the probability of the first event by the conditional probability of the second event given the first event.

$P(A \cap B) = 0$

The probability is 0 because A and B are mutually exclusive and thus, cannot occur at the same time.

$P(A \cup B) = P(A) + P(B) - P(A\cap B)$

Add the probabilities together and less away the intersection to prevent double counting the probability of both A and B occurring.

$P(A \cup B) = P(A) + P(B) - P(A\cap B)$

Add the probabilities together and less away the intersection to prevent double counting the probability of both A and B occurring.

$P(A \cup B) = P(A) + P(B)$

Simply add the probabilities together. Mutually exclusive events do not have probability of them occurring simultaneously.

1. Use the addition rule when you want to find the probability of the union (either one or the other or both) of two mutually exclusive events A and B.
2. Use the multiplication rule when you want to find the probability of the intersection (both events occurring simultaneously) of two independent events A and B.