Probability Top 10 Must Knows (ultimate study guide)

Experimental probability is determined by the results of an experiment. It is calculated by dividing the number of times an event 'A' occurred by the total number of trials. The value of probability always falls between 0 (impossible) and 1 (certain). An example of flipping a coin 10 times is used to illustrate this concept, showing that the more trials conducted, the closer the experimental probability will be to the theoretical probability.

Theoretical probability uses reasoning or math to predict the likelihood of an event without conducting an experiment. It's calculated by dividing the total number of favorable outcomes by the total number of possible outcomes. Examples include rolling a die, drawing a card from a standard deck, and picking marbles from a bag. The concept of complementary probability (the probability of an event not happening) is also introduced.

This section explains probability using Venn diagrams to represent sets. The intersection of sets (denoted as 'and') represents elements common to both sets. The union of sets (denoted as 'or') represents elements in either set or both. Formulas for calculating probabilities involving intersections and unions are provided, with an example based on a survey about sports preferences.

Conditional probability is the likelihood of an event occurring given that another event has already happened. The formula P(B|A) = P(A and B) / P(A) is introduced. A visual representation using Venn diagrams helps to explain how the sample space is narrowed when a condition is given. An example using a table about school preferences and gender illustrates its application.

The multiplication law is used to find the probability of multiple events happening in sequence. For independent events (where one does not affect the other), the probabilities are simply multiplied. For dependent events (where one affects the other), the conditional probability of the second event is used. Examples include rolling a die and flipping a coin (independent), and drawing two kings without replacement from a deck of cards (dependent).

Permutations are ordered arrangements of objects. The number of permutations of n distinct objects is n factorial (n!). For ordered arrangements of n items taken r at a time, the formula P(n,r) = n! / (n-r)! is used. Examples include arranging letters and finding the number of ways people can finish a race, and a lock guessing probability.

Combinations are selections of objects where the order does not matter. The number of combinations of n items taken r at a time is calculated using the formula C(n,r) = n! / (r!(n-r)!). This formula is similar to permutations but includes an additional r! in the denominator to account for the lack of order. Examples include forming groups from a set of people and calculating the probability of a team having a specific number of kids.

Continuous probability distributions model variables that can take on an infinite number of values within an interval. The normal distribution is covered as a key example, characterized by its bell-shaped probability density function (PDF). Key properties include the mean being at the center, and specified percentages of data falling within one, two, and three standard deviations. The concept of z-scores and using a z-score table or calculator functions to find probabilities is explained through an example of smoker lifespans.

A binomial probability distribution describes the probability of a fixed number of successes in a set number of independent trials, each with two possible outcomes (success or 'failure'). The probability mass function (PMF) is given as P(X=k) = C(n,k) * p^k * (1-p)^(n-k). An example of rolling a six-sided die four times and counting the number of threes rolled is used to illustrate this discrete probability distribution. The expected value (mean) of a binomial distribution is also covered.

A geometric probability distribution models the number of trials needed to achieve the first success in an experiment, where trials are independent and have a constant probability of success. The PMF is P(X=k) = (1-p)^(k-1) * p. An example of rolling a pair of dice until doubles are achieved demonstrates how to calculate the probability of the first success occurring on a specific trial or within a certain number of trials. The expected waiting time for the first success is also explained as 1/p.