Events

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Events

Events are certain subsets of the sample space; we will be able to define the probability of an event. For our purposes, we may think of any subset of the sample space as being an event. (In other situations, there are subsets of the sample space that can't be considered as events, and for which no probability can be defined.)
Example: For the experiment of tossing two coins, there are four possibilities in the sample space, since there are four outcomes: HH,HT,TH,TT. HH is the outcome of a head on the first and the second, HT is the outcome of getting a head on the first and tails for the second, TH is the outcome of getting a tail on the first and a heads on the second, and TT is the outcome of getting a tail on both. We think of each of these as being a single outcome of the experiment of tossing the coin twice. Remember that events are (in the examples we will look at) subsets of the sample space. Here are all the possible events for this setup:

{HH,HT,TH,TT}	the whole sample space; "the event that you got either a head or tail on the first toss, and also on the second."
{HH,HT,TH}	the event that you got a head on at least one toss - for example, if I toss a head on the first try and then a tails, this would be written HT, and since the possibility HT is in this set, HT is in the event that we got a head on at least one toss. The events may contain more than one outcome. In this case there are several ways to realize the event "head on at least one toss".
{HH,TH,TT}	the event that you did not get a head followed by a tail
{HH,HT,TT}	the event that you did not get a tail followed by a head
{HT,TH,TT}	the event that you got a tail on one or more tosses
{HH,HT}	the event that you got a heads on the first toss
{HH,TH}	the event that you got a heads on the second toss
{HT,TT}	the event that you got a tails on the second toss
{TH,TT}	the event that you got a tails on the first toss
{HH,TT}	the event that you got the same thing on both tosses
{TH,HT}	the event that you got something different on both tosses or the event that you got one heads
{HH}	the event that you got a heads both times (two heads)
{HT}	the event that you got a heads followed by a tails
{TH}	the event that you got a tails followed by a heads
{TT}	the event that you got a tails both times
{}	the event that you got neither a heads nor a tails on both tries

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