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Events and probability
Before asking “what is the probability?”, ask “what can happen?” A sample space lists the possible outcomes. An event collects the outcomes that answer a particular question.
For two independent fair six-sided dice, the ordered pair (1, 6) is different from (6, 1). There are 6 × 6 = 36 equally likely pairs. The sums are not equally likely: only one pair gives 2, but six give 7.
10 / 36 = 27.78%. Both dice are independent and fair. Conditioning removes incompatible outcomes before you count.
Count the right things
Start with the first-die threshold at 1, so all 36 pairs remain possible. With the sum threshold at 10, the favorable pairs are (4,6), (5,5), (6,4), (5,6), (6,5), and (6,6). The probability is 6/36 = 1/6.
Counting favorable outcomes and dividing by all outcomes works here because the pairs are equally likely. For loaded dice, assign a probability to each pair and add those probabilities instead. “Six ways” alone would no longer determine the answer.
Complements save work
The complement of an event contains every outcome outside it. Their probabilities add to one. To find the chance of at least one six in two rolls, first find the chance of no six: (5/6)². The desired probability is 1 − 25/36 = 11/36.
Make a prediction
Why is twice 1/6 the wrong answer for at least one six?
Explore the answer
It counts (6,6) twice. Adding the probabilities of two events requires subtracting their overlap: 1/6 + 1/6 − 1/36 = 11/36.
Overlap and exclusivity
Two events are mutually exclusive if they cannot happen together. “First die is 1” and “first die is 6” are exclusive. “First die is 6” and “second die is 6” overlap.
Mutually exclusive does not mean independent. Knowing that the first die is 1 rules out its being 6. By contrast, knowing the first die does not change the second die’s probabilities under our independent-roll model.
Try a new question
Change the sum threshold to 8 and count before reading the result. Then use the complement to find the chance of a sum below 8. Both answers must add to one. Next, change the first-die threshold: you have begun conditioning, where the denominator changes.
The same complement strategy makes the birthday problem manageable: it is easier to count no shared birthdays than all possible ways a match can occur.