Probability surfaces in many aspects of daily life. What is the probability that it will rain tomorrow? What is the chance that I will roll a six with these dice? Probability is also a central concept underpinning much of statistics. From the ideas behind simple random sampling, to interpretation of p-values and confidence intervals, probability and probability manipulations provide the building blocks of much of data analysis.
Learning Objectives
For the probability section of the exam, you should be able to manipulate probabilities based on the probability axioms (rules) and understand the difference among types of probabilities (e.g., joint, marginal, conditional). You should be able to make calculations of probability based on the normal distribution.
We have broken down the probability section into the following modules:
This section provides an overview of the key probability concepts that will be covered on the diagnostic exam.
Probability: the likelihood that an event will occur. For example, there is a 50% probability that a fair coin will come up heads on any given flip. Probabilities can be expressed as percents (30%), in decimal form (o.3) or in fractions (3/10). In statistics we most often deal with probability as decimals.
Probability (frequentist): over the long run, the proportion or percentage of time that an event will occur out of all observations. For example, I rolled one die a hundred times. Seventeen times out of a hundred the die showed a value of one. The probability of getting a one is therefore 0.17.
Probability (subjective): a measure of strength of belief. For example, the likelihood that it will storm this evening is 0.7 [p(storm)=0.7].
Random variable: a variable (often denoted X) is a variable whose value is a function of a random process. In other words, the value is determined by chance or a stochastic process (can also think of as an experiment or data generating process). Here is a quick Kahn Academy video on random variables.
Discrete random variable: the random outcomes are countable (finite) and values between these counts can not occur. An example: A random variable, X, takes on the value of one if a coin shows heads, and zero if tails. The expected value or mean (μ) of a discrete random variable is Σxp(x).
Continuous random variable: outcomes and related probabilities are not defined at specific values, but rather over an interval of values. An example: A random variable, X, the weight of an adult blue crab caught from North Creek, may range from 0.5 to 3.0 kg. The probability that an adult weighs between these values is the area under the curve of the probability density function.
Sample Space, Independent and Dependent Events
Sample space (S): the collection of all possible outcomes. The sample space for the roll of a die is:
S={1, 2, 3, 4, 5, 6}
Complement (often denoted as Ac or A with a bar on top) : the probability of the complement of A includes the sum of all probabilities in the sample space that are not A. For example, the probability of the complement of rolling a five on a die (S={1,2,3,4,6}) equals 5/6.
Mutually exclusive events: Two (or more) events that can not occur at the same time. p(A and B) = 0. Example: The Duke’s women’s basketball team can not both win (event A) and lose (event B) a game, therefore p(win and lose) or p(A and B) = 0. We also call these events disjoint.
Independent events: If an event occurring does not alter the probability of another event occurring, we say that these events are independent. For example, if we roll a die twice, getting a three on the first roll does not affect the probability of getting a three on the second roll. Therefore, we can say that these two events are independent.
Dependent events: if an event occurring (A) changes the probability of another event occurring (B), we can say that the probability of event B is dependent on event A. For example, the probability of elevated ground-level ozone concentrations is dependent on the occurrence of a large traffic jam. To learn more about independent and dependent events, please see Khan Academy’s website on dependent probabilities.
