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Probability

Many probability classes start with counting methods such as permutations and combinations. Counting methods are a topic in combinatorics.

If you are studying these topics, see the lessons in combinatorics.

In probability, there is some experiment that has a set of possible outcomes. The set of all possible outcomes is called the sample space. Usually \(\Omega\) is used to represent the sample space.

It is possible to measure the likelihood that the outcome of the experiment will be in some subsets of \(\Omega.\) Each measurable set is called an event, and the set of all the measurable sets is called the event space. The notation for the event space is \(\mathcal{E}\).

Most undergraduate courses do not discuss the event space, and we will not make it a focus in this course.

For the interested reader, the event space \(\mathcal{E}\) must satisfy the following:

- \(\mathcal{E} \subset \mathcal{P}(\Omega)\)
- \(\Omega \in \mathcal{E}\)
- If \(A \in \mathcal{E},\) then \(A^C \in \mathcal{E}.\)
- The union of a countable set of elements of \(\mathcal{E}\) is in \(\mathcal{E}.\) So, if \(A_1, A_2, \dots \in \mathcal{E}\) then \[\bigcup_{i=1}^\infty A_i \in \mathcal{E}.\]

A probability is a measure of the likelihood that an event occurs. The notation for a probability is \(P\).

A probability \(P\) on the sample space \(\Omega\) with event space \(\mathcal{E}\) satisfies the following:

- \(P : \mathcal{E} \rightarrow [0, 1]\)
- \(P(\Omega) = 1\)
- If \(A_1, A_2, \dots \in \mathcal{E}\) is a disjoint sequence of events, meaning \(A_i \cap A_j = \emptyset\) whenever \(i \neq j,\) then \[P\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty P(A_i)\]

Suppose you work for a company that sells 2 products — a cheap one and an expensive one. Your next work task depends on the success of the sales team.

- A. If the sales team sells the expensive product, you need to follow up and thank the customer.
- B. If the sales team sells the cheap product, you need to thank the customer and remind them of the benefits of the expensive product.
- C. If the sales team did not make a sale, but the customer expressed interest, a different team will follow up with the customer. You will not do anything.
- D. If the customer was not interested in any products, they will be left alone. You will not do anything.

There are 10 sales pitches every day, roughly 2 of which have result A and 5 of which have result B.

A customer is meeting with the sales team in 5 minutes. You can represent the likelihood that you will have to do job A or job B with a probability.

There are 4 possible outcomes from the point of view of the sales team which make up the state space, \(\Omega.\) \[\Omega = \{A, B, C, D\}\] You can only measure the likelihood of the outcomes \(A\) and \(B\). If \(C\) or \(D\) happens, you will know that \(A\) and \(B\) did not happen, but you will not know whether \(C\) or \(D\) happened. In this case, you cannot measure whether or not \(C\) happened, thus restricting the event space to what you can measure. Similarly, you cannot measure whether or not \(D\) happened. The event space, based on your knowledge, is \[\mathcal{E} = \{\{A\}, \{B\}, \{C, D\}, \{A, B\}, \{A, C, D\}, \{B, C, D\}, \emptyset, \Omega\}\] The event space is the set of all events for which you can find a probability.

The probability can be approximated by the number of outcomes for an event divided by the total number of outcomes. \begin{align} & P(\{A\}) = 2/10 \\ & P(\{B\}) = 5/10 \\ & P(\{C, D\}) = 3/10 \end{align} With this information, you can find the probability of any event in the event space. For example, \begin{align} P(\{A, C, D\}) & = P(\{A\} \cup \{C, D\}) \\ & = P(\{A\}) + P(\{C, D\}) \\ & = 2/10 + 3/10 \\ & = 5/10 \\ & = 1/2 \end{align}

Quiz:

Define \(\Omega = \{a, b, c, d, e\}.\) The probability is defined by
\[P(a) = 0.3, P(b) = 0.1, P(c) = 0.05, P(d) = 0.25, P(e) = 0.3\]
Define the events \(A = \{a, b\},\) \(B = \{c, d\},\) and \(C = \{a,d,e\}.\)

1. What is \(P(\{a, b, c\})\)?

Unanswered

2. Find \(P(A).\)

Unanswered

3. Compute \(P(B^C).\)

Unanswered

4. Compute \(P(A \cup C).\)

Unanswered