Simple Probability

The probability of an outcome or sample point is a real number, between 0 and 1, that provides a measure of likelihood that the outcome or sample point will actually occur. A sample point that absolutely cannot occur has a probability of 0, while a sample point that will always occur has a probability of 1; all other sample points are assigned a probability based on this relative measure.

The probability of an event A is the summation of the probabilities of all the sample points in A. It is denoted by P(A). If event A is a subset of the sample space S, then 0 £ P(A) £ 1. If A = Æ , then P(A) = P(Æ) = 0; if A = S, then P(A) = P(S) = 1. Otherwise, the value of P(A) is between 0 and 1 exclusive, and there is a positive probability that event A can occur.

EX. When tossing a regular die, one will roll the numbers in the set { 1, 2, 3, 4, 5, 6 } with equal probability. If we denote the event that the number n is rolled by P(n), then

P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 0

Since P(n) has the same value for n Î { 1, 2, 3, 4, 5, 6 },

6 P(n) = 1
P(n) = 1/6 for n Î { 1, 2, 3, 4, 5, 6 }

If event A is the event that an odd number is rolled, then

P(A) = P(1) + P(3) + P(5)
= 1/6 + 1/6 + 1/6
= 1/2

If event B is the event that the number 7 is rolled, then obviously event B will never happen. Thus, event B is the null space, and P(B) = P(Æ) = 0.

On the other hand, if event C is the event that a one-digit number is rolled, then event C will always happen, since the numbers 1 to 6 all have one digit. Thus, P(C) = 1.