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Essential Concepts

• A probability model includes all possible outcomes of a chance experiment and the probabilities associated with those outcomes. A probability model is also known as a probability distribution. For a probability distribution to be valid:
  • The outcomes are random events.
  • All outcomes are assigned a probability. The probabilities are numbers between 0 and 1.
  • The sum of all of the probabilities is 1.
• In a uniform probability model, each outcome has equal probability (for example, rolling a die where each face has a 1/6 probability of occuring)
• For a discrete random variable:
  • The values associated with the random variable of interest are numerical and discrete.
  • All possible values of the random variable are listed in a table or graph with each value having an associated probability greater than or equal to 0 and less than or equal to 1.
  • The sum of all probabilities in the table or graph equals 1.
• A continuous probability distribution is a probability distribution for a continuous random variable (an infinite and uncountable random variable). For all continuous random variables, the probability distribution can be approximated by a smooth curve called a probability density curve. The probabilities of a continuous probability distribution are represented as the area under a density curve.
• A normal distribution is a mathematical model with a smooth bell-shaped curve to describe the bell-shaped data distributions.
• A normal distribution has the following characteristics:
  • [latex]X[/latex] is a continuous random variable.
  • The mean is the center of the distribution which is symmetrical, the left side is a mirror image of the right side centered at the mean.
  • Bell shaped: There is one peak (unimodal) at the mean.
• The standard deviation of a normal distribution, [latex]\sigma[/latex], will change depending on how spread out or flat the curve appears.
• A normal distribution with a mean ([latex]\mu[/latex]) = 0 and a standard deviation ([latex]\sigma[/latex]) = 1 is called the standard normal distribution (or [latex]z[/latex] distribution).
• To compare [latex]x[/latex]-values from different distributions, we can standardize the values into a standard normal distribution by converting the [latex]x[/latex]-values into their respective [latex]z[/latex]-scores.
• When working with probability distributions, we often need to find the probability of values falling in certain regions of the distribution. There are three main types of probability calculations: 
  • Upper tail [latex]P(X>a)[/latex] finds the probability of the top percent, or events that exceed given outcome.
  • Lower tail [latex]P(X
  • Interval find the probability of an outcome occurring between two events. The probability of an outcome between two values can be found as [latex]P(a

Key Equations

[latex]z[/latex]-score

[latex]z=\dfrac{x-\mu}{\sigma}[/latex], where [latex]x[/latex] represents the value of the observation, [latex]\mu[/latex] represents the population mean, [latex]\sigma[/latex] represents the population standard deviation, and [latex]z[/latex] represents the standardized value, or z-score.

Glossary

continuous probability distribution

a probability distribution for a continuous random variable (an infinite and uncountable random variable)

discrete probability distribution

a type of probability distribution that shows all possible values of a discrete random variable (countable or finite outcomes) along with the probabilities associated with those outcomes

interval probability

the probability of observing a value between two specified values

lower tail probability

the probability of observing a value less than a specified value; the area under a probability curve to the left of a specified point

normal distribution

a mathematical model with a smooth bell-shaped curve to describe the bell-shaped data distributions

probability density curve

the probability distribution that can be approximated by a smooth curve

probability model (probability distribution)

all possible outcomes of a chance experiment and the probabilities associated with those outcomes

standard normal distribution

a normal distribution with a mean ([latex]\mu[/latex]) = 0 and a standard deviation ([latex]\sigma[/latex]) = 1

uniform probability

model a probability model where all possible outcomes have equal probability of occurring (e.g., rolling a fair die where each number has probability 1/6)

upper tail probability

the probability of observing a value greater than a specified value; the area under a probability curve to the right of a specified point