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Summary

10.8 Summary (EMCK6)

  • The addition rule (also called the sum rule) for any \(\text{2}\) events, \(A\) and \(B\) is \[P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\] This rule relates the probabilities of \(\text{2}\) events with the probabilities of their union and intersection.

  • The addition rule for \(\text{2}\) mutually exclusive events is \[P(A \text{ or } B) = P(A) + P(B)\] This rule is a special case of the previous rule. Because the events are mutually exclusive, \(P(A \text{ and } B) = 0\).

  • The complementary rule is \[P(\text{not } A) = 1 - P(A)\] This rule is a special case of the previous rule. Since \(A\) and \((\text{not } A)\) are complementary, \(P(A \text{ or } (\text{not } A)) = 1\).

  • The product rule for independent events \(A\) and \(B\) is:

    \[P(A\text{ and } B) = P(A) \times P(B)\]

    If two events \(A\) and \(B\) are dependent then: \[P(A\text{ and } B) \ne P(A) \times P(B)\]

  • Venn diagrams are used to show how events are related to one another. A Venn diagram can be very helpful when doing calculations with probabilities. In a Venn diagram each event is represented by a shape, often a circle or a rectangle. The region inside the shape represents the outcomes included in the event and the region outside the shape represents the outcomes that are not in the event.

  • Tree diagrams are useful for organising and visualising the different possible outcomes of a sequence of events. Each branch in the tree shows an outcome of an event, along with the probability of that outcome. For each possible outcome of the first event, we draw a line where we write down the probability of that outcome and the state of the world if that outcome happened. Then, for each possible outcome of the second event we do the same thing. The probability of a sequence of outcomes is calculated as the product of the probabilities along the branches of the sequence.

  • Two-way contingency tables are a tool for keeping a record of the counts or percentages in a probability problem. Two-way contingency tables are especially helpful for figuring out whether events are dependent or independent.

  • The fundamental counting principle states that if there are \(n(A)\) outcomes for event \(A\) and \(n(B)\) outcomes for event \(B\), then there are \(n(A) \times n(B)\) different possible outcomes for both events.

  • When you have \(n\) objects to choose from and you choose from them \(r\) times, if the number of choices remains the same after each choice, then the total number of possibilities is \[n \times n \times n \ldots \times n \enspace (r \text{ times}) = n^{r}\]

  • The number of arrangements of \(n\) different objects is \[n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 = n!\]

  • For a set of \(n\) objects, of which there are k subsets with repeated objects i.e. \(n_{1}\) are the same, \(n_{2}\) are the same, \(\ldots, n_{k}\) are the same, the number of arrangements are \[\dfrac{n!}{n_{1}! \times n_{2}! \dots n_{k}!}\]