Probability quantifies uncertainty: it assigns a likelihood to an event so that risks, predictions, and decisions can be assessed systematically. It underpins statistical inference and models of uncertain outcomes. section 2.0 section 1.1.4
For equally likely outcomes, probability is the number of favorable outcomes divided by the total number of possible outcomes. Define the event before counting: a product malfunction, for example, can be the outcome of interest even though it is undesirable. Check both the definition of success and the denominator, especially when outcomes involve multiple dice or cards drawn without replacement. section 2.1
Choose a calculation that matches the event:
- Use counting rules, combinations, or permutations to establish how many outcomes qualify and how many are possible. section 2.0
- Multiply probabilities for independent events that must both occur; add probabilities for mutually exclusive alternatives when either can occur. section 3.2
- Distinguish probability from odds: probability compares successes with all outcomes, while odds compare successes with failures. section 2.1.2
Count an event before calculating it
Two fair dice produce 6 × 6 = 36 ordered outcomes. If the first must show 1, 2, or 3 and the second must show 4, 5, or 6, there are 3 × 3 = 9 favorable outcomes. The probability is 9/36, or 25%. Treat (3, 4) and (4, 3) as different outcomes; only the first meets these conditions. section 3.2.3
>>> probability_heads = 1 / 2
>>> probability_heads_percent = probability_heads * 100
>>> print(f'The probability of success equals: '
f'{probability_heads_percent}%')
The probability of success equals: 50.0%
This short Python calculation shows how a 1/2 probability becomes a displayed 50.0%. Multiplying by 100 changes its presentation, not the underlying likelihood. section 2.1.2
The framework extends beyond equally likely game outcomes. Random variables and probability distributions structure uncertain quantities; conditional probability addresses likelihood given information. In a decision tree, probabilities label uncertain branches alongside their payoffs. In Markov analysis, transition calculations can estimate the probabilities of reaching absorbing states. Interpret any result in light of how its outcomes and assumptions were specified. section 1.4.3 section 10.2 section 11.4.2
