Six numbers, two circles, two words people mix up.

Roll one fair die and everything in this chapter is already in front of you: a sample space, a union, an intersection, and the difference between mutually exclusive and independent.

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Six equally likely outcomes, so every probability is just a count:

The rectangle holds all of them — that is what makes $P(S) = 1$.

$A$ is "even", $B$ is "a multiple of 3". Every outcome lands in exactly one of four regions, and 6 is the only one in both.

Shade both circles and the lens goes darker — because the 6 has been counted twice. Take it off once:

Which is the four outcomes 2, 3, 4 and 6, as the picture says.

Independence is a calculation, not a feeling. Multiply and compare:

They match, so $A$ and $B$ are independent: an even score doesn't change the chance of a multiple of 3.

Now take $C$: "the score is 5". $A$ and $C$ cannot both happen, so they are mutually exclusive and $P(A \cap C) = 0$.

But $P(A) \times P(C) = \tfrac{1}{12}$, which is not 0. Knowing the score is 5 drops the chance of "even" from $\tfrac12$ to nothing — the strongest influence there is. Mutually exclusive events can never be independent.

Two circles will take you a long way.

Sample spaces, three-set Venn diagrams, tree diagrams, sampling with and without replacement, and twelve graded exam questions with full worked solutions are waiting.

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