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Total Probability Formula

(conditional probability)


We now learn about the total probability formula, for conditional probability.

Given two events \(A\) and \(B\), such that the probability of \(A\) is affected by whether or not event \(B\) has occurred, then to calculate the probability of event \(A\) occuring we need to consider the following two possible mutually exclusive events:

  • \(A\cap B\): this is the event \(A\) and \(B\), meaning \(A\) occurs and \(B\) occurred

  • \(A\cap B'\): this is the event \(A\) and \(B'\), meaning \(A\) occurs and \(B\) didn't occur
Since these two events are mutually exclusive (cannot happen simultaneously) the probability of \(A\) is given by: \[p\begin{pmatrix} A \end{pmatrix} = p\begin{pmatrix} A \cap B \end{pmatrix} + p\begin{pmatrix} A \cap B' \end{pmatrix} \] Using the multiplicative probability formula, \(p\begin{pmatrix} A \cap B \end{pmatrix} = p\begin{pmatrix} B \end{pmatrix} \times p\begin{pmatrix} A | B \end{pmatrix}\) this turns into the total probability formula:

Total Probability Formula


Formula

Given two events \(A\) and \(B\), if the probability of event \(A\) is affected by whether or not event \(B\) occors then we can calculate the probability of event \(A\) occuring using: \[p(A) = p(B)\times p\begin{pmatrix} A | B \end{pmatrix} + p(B')\times p\begin{pmatrix} A | B' \end{pmatrix} \] This formula is "saying":

For event \(A\) to happen: either the event \(B \cap A\) happened, or \(B' \cap A\) happened, but not both.

Tree Diagram

The total probability formula makes a litle more sense when we read it off a tree diagram.
We do this here, highlighting the key paths:


Looking at this tree diagram, we observe starting from the left:

  • the first part corresponds to the two possibilities: event \(B\) occuring and event "not \(B\)" occurring, that's \(B'\).
  • the second part shows the corresponding two possible events (\(A\) or "not \(A\)", that's \(A'\))following both \(B\) and \(B'\), occuring (so four possibilities in total).
  • On the far right hand side we've added the probabilities of the two paths in which event \(A\) occurs. Those are: \[p\begin{pmatrix} A \cap B \end{pmatrix} = p \begin{pmatrix} A \end{pmatrix} \times p\begin{pmatrix} B | A \end{pmatrix}\]
    and
    \[p\begin{pmatrix} A' \cap B \end{pmatrix} = p \begin{pmatrix} A' \end{pmatrix} \times p\begin{pmatrix} B | A' \end{pmatrix}\]

Tutorial

In the following tutorial we learn how the total probability formula can be derived using a tree diagram as well as watch a worked example. Watch it now.

Exercise


  1. The probability that it will rain during any day in February is 0.2.
    When it rains there is a \(40 \% \) chance that students play football during their afternoon break. On the other hand, when it doesn't rain there is a \(90 \% \) chance that they play.
    What is the probability that students play football on February \(15^{\text{th}}\) ?

  2. Cathy occasionally likes to drink soda. She mainly drinks it if she eats fried chicken on Tuesdays ("special deal day").
    The probability that she eats fried chicken on any Tuesday is \(0.8\). If she eats fried chicken the probability of her drinking soda is \(0.4\), otherwise it is \(0.05\).
    Calculate the probability that Cathy will drink soda next Tuesday.

  3. In a grade \(12\) class, the probability that a students studies Higer Level mathematics (HL) is \(0.3\), altenratively students will study Standard Level Mathematics (SL).
    \(80 \% \) of students studying HL study Two Sciences, whereas only \(40 \% \) of the SL students study Two Sciences.
    A student is selected at random from the class. What is the probability that this student studies Two Sciences?

  4. On any given day there is a \(80 \% \) chance that John has to take an object away from his two year old daughter, Charlotte.
    If he takes an object away from her there is a \(90\% \) chance she cries.
    If he doesn't take an object away from her there is still a \(30\% \) chance that she cries that day.
    What is the probability that Charlotte cries tomorrow?

Answers Without Working

  1. Defining the even \(F\) as: grade \(12\) students play football on February \(15^{\text{th}}\), we find: \[p\begin{pmatrix} F \end{pmatrix} = 0.8\]

  2. Defining the even \(S\) as: Cathy drinks a soda next Tuesday: \[p\begin{pmatrix} S \end{pmatrix} = 0.33\]

  3. Defining the even \(TS\) as: the student studies two sciences: \[p\begin{pmatrix} TS \end{pmatrix} = 0.52\]

  4. Defining the even \(C\) as: Charlotte cries tomorrow: \[p\begin{pmatrix} C \end{pmatrix} = 0.78\]