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People often make the following definition:

If an action or experiment has a finite set of possible outcomes that can be deemed equally likely, then the probability or chance of any one particular outcome occurring is the number

\(\dfrac{1}{\text{Total number of outcomes}}\).

For example, in rolling a die there are six possible outcomes—rolling 1, 2, 3, 4, 5, or 6 —all usually deemed equally likely. Then we say:

probability of rolling a three \(=\dfrac{1}{6}\),

probability of rolling a six \(=\dfrac{1}{6}\),

and so on.

Some jargon:

Definition: The set of all possible outcomes of an experiment is called its sample space.

For example, the action of tossing a coin has sample space \(\{\text{heads},\text{tails})\), the action of rolling a die has sample space \(\{1, 2, 3, 4, 5, 6\}\), and the action of ascertaining someone’s age in years has sample space \(\{0,1,2,3,\ldots, 120?\}\).

Definition: An event for some experiment or action is a subset of its sample space.

For example, in the action of rolling a die, an event could be \(\{2,4,6\}\) (rolling an even number), or \(\{1\}\) (rolling a one), or \(\{1,2,3,4,5,6\}\) (rolling any number), or  \(\{\}\) (rolling, but failing to have a number show).

It is natural to extend the basic definition given above as follows

Definition: For an experiment or action with sample space \(S\), we define the probability of a particular event \(E\) occurring to be:

\(p\left(E\right)=\dfrac{\text{size of } E}{\text{size of }S}\).

Here we are assuming that the sample space has just a finite number of elements and that every single outcome is “equally likely.”

For example, in rolling a die

\(p\left(\text{even}\right)=\dfrac{\text{size of } \{2,4,6\}}{\text{size of }\{1,2,3,4,5,6\}}=\dfrac{3}{6}=\dfrac{1}{2}\).

Also

\(p\left(\{3\}\right)=\dfrac{1}{6}\),

\(p\left(\{1,2,4,5\}\right)=\dfrac{}{6}=\dfrac{2}{3}\),

\(p\left(\text{rolling a }7\right)=\dfrac{0}{6}=0\),

\(p\left(\text{rolling any number}\right)=\dfrac{6}{6}=1\).

Example: The following table shows the number of students at a certain school by grade and their preferred flavor: butterscotch or caramel.

a) If a student is chosen at random, what are the chances the student would prefer caramel over butterscotch?

b) If a student prefers butterscotch over caramel, what are the chances that the students is in tenth grade?

Answer: a)  \(\dfrac{397}{512}\approx 78\%\)  b)  \(\dfrac{110}{512}\approx 21\%\) .

Comment: Suppose a game with a finite number of possible outcomes has \(a\) outcomes deemed “favorable” with the remaining \(b\) deemed “unfavorable.” Then people sometimes call the ratio \(a:b\) the odds in favor of winning. For example, in rolling a die, the odds in favor of rolling a 6  are \(1:5\). The odds against rolling a 5 or a 6 are \(4:2\) (which could be reduced to \(2:1\)). In a horse race if the odds against a horse are \(7:2\) , this means that bookies believe that the horse has only a \(\frac{2}{9}\) chance of winning.

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