Examples

Coin flipping

Flip a fair coin two times and record both results.

• Outcomes: sequences, like $HH$ or $TH$.

• Sample space: all possible sequences, i.e. the set $S=\{HH,HT,TH,TT\}$.

• Events: for example:

  • $A=\{HH,HT\}=\text{“first was heads”}$
  • $B=\{HT,TH\}=\text{“exactly one heads”}$
  • $C=\{HT,TH,HH\}=\text{“at least one heads”}$

With this setup, we may combine events in various ways to generate other events:

Complex events: for example: $A\cap B=\{HT\}$, or in words:

$$\text{“first was heads”}\text{AND}\text{“exactly one heads”}=\text{“heads-then-tails”}$$

Notice that the last one is a complete description, namely the outcome $HT$. $A\cup B=\{HH,HT,TH\}$, or in words:

$$\begin{array}{c}\text{“first was heads”}\text{OR}\text{“exactly one heads”}\\ =\text{“starts with heads, else it’s tails-then-heads”}\end{array}$$

Coin flipping: counting subsets

Flip a fair coin five times and record the results.

How many elements are in the sample space? (How big is $S$?) How many events are there? (How big is $ℱ$?)

Solution

There are $2^5=32$ possible sequences, so $|S|=32$.

To count the number of possible subsets, consider that we have 32 distinct items, and a subset is uniquely determined by the binary information – for each item – of whether it is in or out. Thus there are $2^{32}$ possibilities. So $|ℱ|=2^{32}$.