2.2.5 Equally likely outcomes

In many experiments consisting of $N$ outcomes, it is reasonable to assign equal probabilities to all $N$ simple events. These include such obvious examples as

• tossing a fair coin or fair die once or twice (or any fixed number of times),
• or selecting one or several cards from a well-shuffled deck of 52. With $p=P\left(E_i\right)$ for every $i$,

$$1=\sum _{i=1}^{N}P\left(E_i\right)=\sum _{i=1}^{N}p=p\cdot N$$

so $p=\frac{1}{N}$. That is,

• if there are $N$ equally likely outcomes,
• the probability for each is $1/N$ .

Now consider an event $A$ ,

• with $N(A)$ denoting the number of outcomes contained in $A$ . Then

$$P\left(A\right)=\sum _{E_i\text{ in }A}P\left(E_i\right)=\sum _{E_i\text{ in }A}\frac{1}{N}=\frac{N\left(A\right)}{N}$$

Thus when outcomes are equally likely, computing probabilities reduces to

• counting:
  • both the number of outcomes $N\left(A\right)$ in $A$
  • and the number of outcomes $N$ in $\mathcal{S}$,
• and form their ratio.

EX 2.16 equally likely outcomes