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What is a random event?

Module by: Brandon Hodgson

What is a random event?

An event is a set of outcomes (a subset of the sample space) to which a probability is assigned (Wikipedia 2006a). Typically, any subset of the sample space is an event (i.e. all elements of the power set of the sample space are events) (Wikipedia 2006a).

Example 1

This example is extracted from (Wikipedia 2006a). If we assemble a deck of 52 playing cards with two jokers, and draw a single card from the deck, then the sample space is a 54-element set, as each individual card is a possible outcome. An event, however, is any subset of the sample space, including any single-element set (an elementary event, of which there are 54, representing the 54 possible cards drawn from the deck), the empty set (which is defined to have probability zero) and the entire set of 54 cards, the sample space itself (which is defined to have probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential events include:

  • "Red and black at the same time without being a joker" (0 elements),
  • "The 5 of Hearts" (1 element),
  • "A King" (4 elements),
  • "A Face card" (12 elements),
  • "A Spade" (13 elements),
  • "A Face card or a red suit" (32 elements),
  • "A card" (54 elements).

Since all events are sets, they are usually written as sets (e.g. {1, 2, 3}), and represented graphically using Venn diagrams (illustrations used in the branch of mathematics known as set theory)(Wikipedia 2006a, b). Venn diagrams are particularly useful for representing events because the probability of the event can be identified with the ratio of the area of the event and the area of the sample space (Wikipedia 2006a). An example of a Venn diagram is shown in figure 1 (Wikipedia 2006).

Figure 1
Figure 1 (150px-Venn_A_subset_B)
(Wikipedia 2006)

Figure 1: A Venn diagram of an event. B is the sample space and A is an event. By the ratio of their areas, the probability of A is approximately 0.4 (Wikipedia 2006).

Exercise 1

When sending packets of information through a channel, does receiving erroneous packet qualifies as a random event?

Solution 1

Yes. One cannot predict when exactly a packet of information will be distorted along the channel. Thus randomly, one bit in the information packet may be received wrongly.

References:

Ali M. "Probabilistic Sytems Analyses: Part I," University of the Witwatersrand, 2000.

Wikipedia. "Event (probability theory)," Wikimedia Foundation Inc, http://en.wikipedia.org/wiki/Event_%28probability_theory%29, Last accessed 17 February 2006.

Wikipedia. "Venn diagram," Wikimedia Foundation Inc, http://en.wikipedia.org/wiki/Venn_diagram, Last accessed 17 February 2006.

Author of assignment 3: Brandon Hodgson

Author of assignment 1: Mphekwane (Eddie) Mamahlodi

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