Probability foundations · Definition
Classical probability
Before you start you need to know
It is the oldest model and the first one used in practice: dice, coins, cards and draws where nothing favors one outcome over another.
Definition
The classical probability \(P(A)\) of an event \(A\), when the sample space \(\Omega\) is finite with equally likely outcomes, is the ratio of favorable to possible outcomes.
If \(\Omega\) is finite and its \(n\) outcomes are equally likely, each gets \(\frac1n\) and, by additivity, an event with \(k\) outcomes has probability \(\frac kn\). Computing a probability then reduces to counting: how many favorable cases and how many possible ones.
Guiding example: when two dice are rolled, \(\Omega\) has \(36\) ordered pairs. The sum \(7\) occurs in \(6\) of them, so its probability is \(\frac16\). Counting ordered pairs matters: if the sums \(2,\dots,12\) were used as outcomes, one would conclude \(\frac1{11}\), but those sums are not equally likely.
When direct counting is long, use the complement: the probability of “at least one 6” with two dice is \(1-\frac{25}{36}=\frac{11}{36}\). For large spaces, such as card hands, counting relies on the product rule, permutations and combinations, which we will see later.
The formula holds only if the symmetry of the experiment justifies equal weights; it is not a universal definition. It also fails on infinite spaces and for a loaded die. Relaxing that condition leads to the general probability measure and to the frequentist interpretation.
Think about it
Two coins are tossed. Why is it wrong to say that getting “one head” has probability \(\frac13\) because there are three possible head counts?
Because 0, 1 and 2 heads are not equally likely: “one head” groups two sequences (HT and TH) out of the four equally likely ones, so it has probability \(\frac12\).
Can classical probability be applied to “it rains tomorrow”, with outcomes yes and no?
No. The two outcomes need not be equally likely, and no symmetry justifies it.
Exercises 10
- A fair die is rolled. What is the probability of getting a number greater than 4?
- A fair coin is tossed three times. What is the probability of getting exactly two heads?
- Two fair dice are rolled. What is the probability that the sum is 7?
- One card is drawn at random from a standard 52-card deck. What is the probability that it is a heart?
- An urn has 5 red balls and 3 blue balls. One ball is drawn at random. What is the probability that it is blue?
- Two fair dice are rolled. What is the probability that the sum is at least 9?
- Two fair dice are rolled. What is the probability of getting at least one 6?
- Two cards are drawn at random, without replacement, from a 52-card deck. What is the probability that both are aces?
- Two fair coins are tossed and someone reasons with \(\Omega=\{0,1,2\}\) heads: “\(P(\text{one head})=\frac13\)”. What is the mistake?
- From a group of 4 women and 2 men, a committee of 2 people is chosen at random. What is the probability that it includes at least one woman?
References
- Joseph K. Blitzstein, Jessica Hwang, Introduction to Probability, 2nd edition, CRC Press, 2019. §1.3
- Sheldon Ross, A First Course in Probability, 10th edition, Pearson, 2019. §2.5
- Larry Wasserman, All of Statistics, 1st edition, Springer, 2004. §1.4