Sample spaces, the complement, and what "expected" actually means when you cannot have 12.8 absentees.
The probability of an event is a counting argument: how many outcomes give you what you want, out of how many there are altogether. Whether the world agrees is a separate question. Roll the dice and find out.
At 60 rolls the two can still be a long way apart. That is not the theory being wrong.
Two dice have 36 equally likely outcomes, not 11. The totals run from 2 to 12, but they are not equally likely, because there are six ways to make 7 and only one way to make 2.
| Total | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
P(A) = n(A)n(U), where U is the sample space, only when the outcomes are equally likely. The 36 pairs are. The 11 totals are not, and treating them as equally likely is the classic way to get 1⁄11 instead of 1⁄6.
"At least one" is the signal to use the complement. Counting the ways to get at least one six directly means adding several cases; counting the ways to get none is a single calculation.
An expected number does not have to be possible. With 128 students and P = 0.1 the expected number absent is 12.8, and nobody is 0.8 of a student. It is a long-run average, not a prediction of any particular day, and rounding it to 13 throws away information.
Theoretical probability comes from counting outcomes. Relative frequency comes from doing it and recording what happened. They agree in the long run and they are not the same thing, which is exactly what the simulation above shows: push the rolls up and the gap closes, slowly and unevenly.
1. Two fair dice are rolled. What is the probability the total is 7? Give it to 3 decimal places.
2. A school has 200 students and the probability any one of them is absent is 0.15. How many absences are expected?
3. Why is P(total of 7) not 111?
Stating the sample space, or showing the grid. If the outcomes you count are not equally likely, every probability after that is wrong.
Using the complement when the question says "at least". It is shorter and far less error-prone than adding cases.
Leaving an expected number as a decimal. 12.8 is the answer; 13 is a rounded version of it and sometimes loses the mark.
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