Notes

Expected Value: We can find an expected value by multiplying each numerical outcome by the probability of that outcome, and then summing those products together.

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The Rule of Product: When calculating the probability that multiple independent events will all occur, the probabilities are multiplied.

Example: What is the chance of rolling a sum of 9 followed by a sum of 10 on a pair of traditional dice? Solution: The probability of a sum of 9 is 4/36 . The probability of a sum of 10 is 3/36. The overall probability of a 9 followed by a 10 is 4/36 * 3/36.

Note: Because two die rolls do not affect each other (they are independent), multiplication is allowed. In games where probabilities can change between rounds or where one event might affect another in any way, we'll need more advanced strategies. We'll get into those advanced strategies later. Two events are independent if the probability of one of them occurring does not affect the probability of the other occurring.

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The Rule of Sum: the probability the event as a whole will occur is a sum of the probabilities of each individual part, as long as the parts cannot occur at the same time.

A good rule of thumb (given independence) is the Rule of Product applies when an event AND another event occur, while the Rule of Sum applies when an event OR another event occurs.

Example: What is the chance of rolling a 1, 2, 3, or 4 on a single traditional die?

A 1, 2, 3 or 4 occurs on a single throw with chance 1/6, and each part is independent, so the chance overall is the sum of all four probabilities: 1/6 + 1/6 + 1/6 +1/6

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If you are adding up two unknown quantities (for example the values of two dice), the expected value of their sum equals the sum of their expected values.

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Practice Linearity of Expectation and above

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
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`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$