Probability, Random Variables, and Probability Distributions · ⁨الاحتمالات، المتغيرات العشوائية، والتوزيعات الاحتمالية⁩

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.F: Identify questions suggested by patterns in data. [Skill 1.A]

• VAR-1.F.1 Patterns in data do not necessarily mean that variation is not random.

Enduring Understanding (UNC-2): Simulation allows us to anticipate patterns in data.

Learning Objective UNC-2.A: Estimate probabilities using simulation. [Skill 3.A]

• UNC-2.A.1 A random process generates results that are determined by chance.
• UNC-2.A.2 An outcome is the result of a trial of a random process.
• UNC-2.A.3 An event is a collection of outcomes.
• UNC-2.A.4 Simulation is a way to model random events, such that simulated outcomes closely match real-world outcomes. All possible outcomes are associated with a value to be determined by chance. Record the counts of simulated outcomes and the count total.
• UNC-2.A.5 The relative frequency of an outcome or event in simulated or empirical data can be used to estimate the probability of that outcome or event.
• UNC-2.A.6 The law of large numbers states that simulated (empirical) probabilities tend to get closer to the true probability as the number of trials increases.
  • Illustrative examples for UNC-2.A:
    • An outcome: Rolling a particular value on a six-sided number cube is one of six possible outcomes.
    • An event: When rolling two six-sided number cubes, an event would be a sum of seven. The corresponding collection of outcomes would be $(1, 6)$, $(2, 5)$, $(3, 4)$, $(4, 3)$, $(5, 2)$, and $(6, 1)$, where the ordered pairs indicate (face value on one cube, face value on the other cube).

Enduring Understanding (VAR-4): The likelihood of a random event can be quantified.

Learning Objective VAR-4.A: Calculate probabilities for events and their complements. [Skill 3.A]

• VAR-4.A.1 The sample space of a random process is the set of all possible non-overlapping outcomes.
• VAR-4.A.2 If all outcomes in the sample space are equally likely, then the probability an event E will occur is defined as the fraction: $\dfrac{\text{number of outcomes in event E}}{\text{total number of outcomes in sample space}}$
• VAR-4.A.3 The probability of an event is a number between 0 and 1, inclusive.
• VAR-4.A.4 The probability of the complement of an event E, $E'$ or $E^{C}$, (i.e., not E) is equal to $1 - P(E)$.

Learning Objective VAR-4.B: Interpret probabilities for events. [Skill 4.B]

• VAR-4.B.1 Probabilities of events in repeatable situations can be interpreted as the relative frequency with which the event will occur in the long run.

Enduring Understanding (VAR-4): The likelihood of a random event can be quantified.

Learning Objective VAR-4.C: Explain why two events are (or are not) mutually exclusive. [Skill 4.B]

• VAR-4.C.1 The probability that events $A$ and $B$ both will occur, sometimes called the joint probability, is the probability of the intersection of $A$ and $B$, denoted $P(A \cap B)$.
• VAR-4.C.2 Two events are mutually exclusive or disjoint if they cannot occur at the same time. So $P(A \cap B) = 0$.

Enduring Understanding (VAR-4): The likelihood of a random event can be quantified.

Learning Objective VAR-4.D: Calculate conditional probabilities. [Skill 3.A]

• VAR-4.D.1 The probability that event $A$ will occur given that event $B$ has occurred is called a conditional probability and denoted $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$.
• VAR-4.D.2 The multiplication rule states that the probability that events $A$ and $B$ both will occur is equal to the probability that event $A$ will occur multiplied by the probability that event $B$ will occur, given that $A$ has occurred. This is denoted $P(A \cap B) = P(A) \cdot P(B \mid A)$.

Enduring Understanding (VAR-4): The likelihood of a random event can be quantified.

Learning Objective VAR-4.E: Calculate probabilities for independent events and for the union of two events. [Skill 3.A]

• VAR-4.E.1 Events $A$ and $B$ are independent if, and only if, knowing whether event $A$ has occurred (or will occur) does not change the probability that event $B$ will occur.
• VAR-4.E.2 If, and only if, events $A$ and $B$ are independent, then $P(A \mid B) = P(A)$, $P(B \mid A) = P(B)$, and $P(A \cap B) = P(A) \cdot P(B)$.
• VAR-4.E.3 The probability that event $A$ or event $B$ (or both) will occur is the probability of the union of $A$ and $B$, denoted $P(A \cup B)$.
• VAR-4.E.4 The addition rule states that the probability that event $A$ or event $B$ or both will occur is equal to the probability that event $A$ will occur plus the probability that event $B$ will occur minus the probability that both events $A$ and $B$ will occur. This is denoted $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Enduring Understanding (VAR-5): Probability distributions may be used to model variation in populations.

Learning Objective VAR-5.A: Represent the probability distribution for a discrete random variable. [Skill 2.B]

• VAR-5.A.1 The values of a random variable are the numerical outcomes of random behavior.
• VAR-5.A.2 A discrete random variable is a variable that can only take a countable number of values. Each value has a probability associated with it. The sum of the probabilities over all of the possible values must be 1.
• VAR-5.A.3 A probability distribution can be represented as a graph, table, or function showing the probabilities associated with values of a random variable.
• VAR-5.A.4 A cumulative probability distribution can be represented as a table or function showing the probability of being less than or equal to each value of the random variable.
  • Illustrative examples for VAR-5.A: Outcomes of trials of a random process:
    • The sum of the outcomes for rolling two dice
    • The number of puppies in a randomly selected litter for a certain breed of dog

Learning Objective VAR-5.B: Interpret a probability distribution. [Skill 4.B]

• VAR-5.B.1 An interpretation of a probability distribution provides information about the shape, center, and spread of a population and allows one to make conclusions about the population of interest.

Enduring Understanding (VAR-5): Probability distributions may be used to model variation in populations.

Learning Objective VAR-5.C: Calculate parameters for a discrete random variable. [Skill 3.B]

• VAR-5.C.1 A numerical value measuring a characteristic of a population or the distribution of a random variable is known as a parameter, which is a single, fixed value.
• VAR-5.C.2 The mean, or expected value, for a discrete random variable $X$ is $\mu_X = \sum x_i \cdot P(x_i)$.
• VAR-5.C.3 The standard deviation for a discrete random variable $X$ is $\sigma_X = \sqrt{\sum (x_i - \mu_x)^2 \cdot P(x_i)}$.

Learning Objective VAR-5.D: Interpret parameters for a discrete random variable. [Skill 4.B]

• VAR-5.D.1 Parameters for a discrete random variable should be interpreted using appropriate units and within the context of a specific population.

Enduring Understanding (VAR-5): Probability distributions may be used to model variation in populations.

Learning Objective VAR-5.E: Calculate parameters for linear combinations of random variables. [Skill 3.B]

• VAR-5.E.1 For random variables $X$ and $Y$ and real numbers $a$ and $b$, the mean of $aX + bY$ is $a\mu_x + b\mu_y$.
• VAR-5.E.2 Two random variables are independent if knowing information about one of them does not change the probability distribution of the other.
• VAR-5.E.3 For independent random variables $X$ and $Y$ and real numbers $a$ and $b$, the mean of $aX + bY$ is $a\mu_x + b\mu_y$, and the variance of $aX + bY$ is $a^2\sigma^2_x + b^2\sigma^2_y$.

Learning Objective VAR-5.F: Describe the effects of linear transformations of parameters of random variables. [Skill 3.C]

• VAR-5.F.1 For $Y = a + bX$, the probability distribution of the transformed random variable, $Y$, has the same shape as the probability distribution for $X$, so long as $a > 0$ and $b > 0$. The mean of $Y$ is $\mu_y = a + b\mu_x$. The standard deviation of $Y$ is $\sigma_y = |b|\sigma_x$.

Enduring Understanding (UNC-3): Probabilistic reasoning allows us to anticipate patterns in data.

Learning Objective UNC-3.A: Estimate probabilities of binomial random variables using data from a simulation. [Skill 3.A]

• UNC-3.A.1 A probability distribution can be constructed using the rules of probability or estimated with a simulation using random number generators.
• UNC-3.A.2 A binomial random variable, $X$, counts the number of successes in $n$ repeated independent trials, each trial having two possible outcomes (success or failure), with the probability of success $p$ and the probability of failure $1 - p$.

Learning Objective UNC-3.B: Calculate probabilities for a binomial distribution. [Skill 3.A]

• UNC-3.B.1 The probability that a binomial random variable, $X$, has exactly $x$ successes for $n$ independent trials, when the probability of success is $p$, is calculated as $P(X = x) = \binom{n}{x} p^x (1 - p)^{n-x}, x = 0, 1, 2, \ldots, n$. This is the binomial probability function.

Enduring Understanding (UNC-3): Probabilistic reasoning allows us to anticipate patterns in data.

Learning Objective UNC-3.C: Calculate parameters for a binomial distribution. [Skill 3.B]

• UNC-3.C.1 If a random variable is binomial, its mean, $\mu_x$, is $np$ and its standard deviation, $\sigma_x$, is $\sqrt{np(1 - p)}$.

Learning Objective UNC-3.D: Interpret probabilities and parameters for a binomial distribution. [Skill 4.B]

• UNC-3.D.1 Probabilities and parameters for a binomial distribution should be interpreted using appropriate units and within the context of a specific population or situation.

Enduring Understanding (UNC-3): Probabilistic reasoning allows us to anticipate patterns in data.

Learning Objective UNC-3.E: Calculate probabilities for geometric random variables. [Skill 3.A]

• UNC-3.E.1 For a sequence of independent trials, a geometric random variable, $X$, gives the number of the trial on which the first success occurs. Each trial has two possible outcomes (success or failure) with the probability of success $p$ and the probability of failure $1 - p$.
• UNC-3.E.2 The probability that the first success for repeated independent trials with probability of success $p$ occurs on trial $x$ is calculated as $P(X = x) = (1 - p)^{x-1} p, x = 1, 2, 3, \ldots$. This is the geometric probability function.

Learning Objective UNC-3.F: Calculate parameters of a geometric distribution. [Skill 3.B]

• UNC-3.F.1 If a random variable is geometric, its mean, $\mu_x$, is $\dfrac{1}{p}$ and its standard deviation, $\sigma_x$, is $\dfrac{\sqrt{(1 - p)}}{p}$.

Learning Objective UNC-3.G: Interpret probabilities and parameters for a geometric distribution. [Skill 4.B]

• UNC-3.G.1 Probabilities and parameters for a geometric distribution should be interpreted using appropriate units and within the context of a specific population or situation.