Basic Probability: Calculating Chances in Everyday Life

Probability is the numerical measure of the chance that an event will occur. It ranges from 0 (impossible event) to 1 (certain event) and can be expressed as a fraction, decimal, or percentage. The basic formula is: P(A) = number of favorable outcomes ÷ total number of possible outcomes.

The set of all possible outcomes is called the sample space. When rolling a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. The event 'rolling an even number' corresponds to the subset {2, 4, 6}, so P(even) = 3/6 = 1/2 = 50%.

The probability that an event does NOT occur is called its complementary probability. The formula is simple: P(not A) = 1 − P(A). If the chance of rain tomorrow is 30%, the chance of no rain is 70%.

This concept is very useful when calculating the probability of the opposite event is easier. What is the probability of getting at least one 6 when rolling a die twice? It is easier to calculate P(no 6) = (5/6) × (5/6) = 25/36 and then P(at least one 6) = 1 − 25/36 = 11/36.

Two events are mutually exclusive when they cannot occur at the same time. Getting heads and getting tails on the same coin flip is impossible — these events are exclusive. For mutually exclusive events, P(A or B) = P(A) + P(B). The probability of rolling a 1 or a 6 on a die is 1/6 + 1/6 = 2/6 = 1/3.

Independent events are those where the outcome of one does not affect the other. Flipping a coin and then rolling a die are independent events. For independent events, P(A and B) = P(A) × P(B). The chance of getting heads and then rolling a 4 is 1/2 × 1/6 = 1/12.

Confusing these concepts leads to classic errors. The 'gambler's fallacy' is thinking that after five heads in a row, tails is 'overdue' — but since the flips are independent, the probability of tails on the sixth flip remains 1/2.