Outcomes and probability

Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain).

Sample space and events

An experiment has possible outcomes. The set of all of them is the sample space. An event is a set of outcomes we care about.

Rolling two dice: each die shows 1 to 6, so there are 6 × 6 = 36 outcomes, all equally likely. The event "the sum is 7" contains (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1).

When every outcome is equally likely:

P(event) = favourable outcomes ÷ all outcomes

So P(sum is 7) = 6 ÷ 36 = 1/6. The example counts the outcomes instead of trusting our memory.

from fractions import Fraction

count = 0
for a in range(1, 7):
    for b in range(1, 7):
        if a + b == 7:
            count += 1
print("ways to roll 7:", count)
print("P(sum = 7) =", Fraction(count, 36))

Python's Fraction keeps the answer exact: 1/6, not 0.16666….

Common mistakes

• Counting outcomes that are not equally likely. The sums 2 to 12 are 11 results, but P(sum is 7) is not 1/11: a 7 can happen six ways, a 2 only one way. Count the equally likely outcomes (the 36 pairs), not the results.
• Forgetting that (2, 5) and (5, 2) are different outcomes: the two dice are different dice.

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