What is the expected outcome of rolling a fair six-sided die once?

When rolling a fair six-sided die, each side has an equal probability of landing face up. The sides of the die are numbered from 1 to 6. To find the expected outcome, you calculate the average of all possible outcomes, weighted by their probabilities.

The expected value (E) is calculated using the formula:

[ E = \sum (x_i \cdot P(x_i)) ]

where ( x_i ) represents the outcome and ( P(x_i) ) is the probability of that outcome. For a fair six-sided die, each outcome (1, 2, 3, 4, 5, 6) has an equal probability of ( \frac{1}{6} ).

Calculating the expected value involves the following steps:

1. Multiply each outcome by its probability:

• 1 × ( \frac{1}{6} ) = ( \frac{1}{6} )

• 2 × ( \frac{1}{6} ) = ( \frac{2}{6} )

• 3 × ( \frac{1}{6} ) = ( \frac{3}{6} )

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