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Learning intentions:
In this section we will examine:
- The concept of independent events.
- Equivalent conditions of independence.
Independent events
Two events A and B are independent if the occurrence of one event has no effect/influence on the probability of the other event occurring; that is, if
This rule will always be true for two independent events. An example of two independent events is 'rolling a 4 on a die' and 'flipping a tail on a coin'.
Calculating the probability of the intersection
When two events are independent, the probability of their intersection is given by:
Equivalent conditions of independence
For two events, A and B, with Pr(A) ≠ 0 and Pr(B) ≠ 0, the following three statements are all equivalent conditions for independence:
- Pr(A|B) = Pr(A)
- Pr(B|A) = Pr(B)
- Pr(A ∩ B) = Pr(A) × Pr(B)
If any one of these statements is true for two events, all three statements will be true.
In the special case that Pr(A) = 0 or Pr(B) = 0, the condition Pr(A ∩ B) = Pr(A) × Pr(B) still hold true as both sides will be equal to zero. Therefore, we can say that A and B are independent as one sufficient condition is met.
13D - Exercises:
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Success criteria:
You will be successful if you can:
- Recognise and give examples of independent events.
- Use the three equivalent conditions of independence to determine if two events are independent.
- Apply rules and formulas of independent events to solve problems.
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