Question

a fair die is a cube with number 1 through 6 on the sides, represented as painted spots. if a fair die is rolled, what is the probability of number 6 landing face up?

with a fair die, the probability of rolling any number 1 through 6 is the same.
if \\(p(1)\\) represents the probability of rolling a 1, \\(p(2)\\) the probability of rolling a 2, and so forth, what is the value of: \\(p(1) + p(2) + p(3) + p(4) + p(5) + p(6)\\)?

Explanation:

Identify the sample space and individual probabilities

The sample space \(S\) for rolling a fair six-sided die is:

$$ S = \{1, 2, 3, 4, 5, 6\} $$

Since the die is fair, each outcome is equally likely:

$$ P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = \frac{1}{6} $$

Sum the probabilities of all outcomes

The sum of the probabilities of all mutually exclusive outcomes in a sample space must equal 1:

$$ P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = 1 $$

Answer:

With a fair die, the probability of rolling any number 1 through 6 is the same. If \(P(1)\) represents the probability of rolling a 1, \(P(2)\) the probability of rolling a 2, and so forth, what is the value of: \(P(1) + P(2) + P(3) + P(4) + P(5) + P(6)\)? <blank>1</blank>