http://www.ck12.org Chapter 3. An Introduction to Probability
Now let us return to the concept of probability and relate it to the concepts that we have just studied. You may
be familiar with the meaning of probability and may have used the term as a synonym with informal words like
“chance” and “odds.” For the time being, we will begin our treatment of probability using these informal concepts
and then later, we will solidify these meanings into formal mathematical definitions.
As you probably know from your previous math courses, if you toss a fair coin, the chance of getting a tail(T)is
the same as the chance of getting a head(H). Thus we say that the probability of observing a head is 50% (or 0.5)
and the probability of observing a tail is also 50%. We also say sometimes “the odds are 50−50.”
The probability,P, of an outcome, A, always falls somewhere between two extremes: 0 (or 0%), which means the
outcome is an impossible event and 1 (or 100%) represents an outcome that is guaranteed to happen. These two
extremes are generally not seen in real life situations. Most outcomes have probabilities somewhere in between.
Property 1
0 ≤P(A)≤ 1 For any eventA
The probability of an eventAranges between 0 (impossible) and 1 (always).
In addition, the probabilities of possible outcomes of an event must all add up to 1. This 1 represents a certainty that
one of the outcomes must happen. For example, tossing a coin will produce either a head or a tail. Each of these
two outcomes has a probability of 50%, or 1/2. However, thetotal probabilities of the coin to land head or tail is
1 / 2 + 1 / 2 = 1.
Property 2
∑
all outcomesP(A) = 1
The sum of the probabilities of all possible outcomes must add up to 1.
Notice that tossing a coin or throwing a dice results in outcomes that are all equally probable, that is, each outcome
has the same probability as the other outcome in the same sample space. Getting a head or a tail from tossing a coin
produces equal probability for each outcome, 50%. Throwing a die also has 6 possible outcomes but they all have
the same probability, 1/6. We refer to this kind of probability as theclassical probability. It is the simplest kind
of probability. Later in this lesson, we will deal with situations where each outcome in a given sample space has
different probability.
Probability is usually denoted byPand the respective elements of the sample space (the outcomes) are denoted by
A,B,C,etc. The mathematical notation that indicates that the outcomeAhappens isP(A). We use the following
formula to calculate the probability of an outcome to occur:
P(A) =
The number of outcomes for A to occur
The size of the sample spaceThe following examples show you how to use this formula.
Example:
When tossing two coins, what is the probability of getting head-head(HH)? Is the probability classical?
Solution:
Since there are 4 elements (outcomes) in the set of sample space:{HH,HT,T H,T T}, its size then is 4. Further,
there is only 1HHoutcome to occur. Using the formula above,
P(A) =
The number of outcomes for HH to occur
The size of the sample space