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You roll a 6-sided die.

What is P(3 or factor of 54)?

Simplify your answer and write it as a fraction or whole number.

1 answer

To find \( P(3 \text{ or factor of } 54) \), we first identify the favorable outcomes.

  1. Identify the possible outcomes of rolling a 6-sided die: The die has outcomes {1, 2, 3, 4, 5, 6}.

  2. Identify the outcomes for "3 or factor of 54":

    • The outcome "3" corresponds directly to the face value 3 on the die.
    • Next, we find the factors of 54. The factors of 54 are: \[ 1, 2, 3, 6, 9, 18, 27, 54 \]
    • From the outcomes of the die, the relevant factors of 54 are {1, 2, 3, 6}.

  3. Combine the outcomes:

    • The outcomes we care about, considering the "or" condition, are:
      • From the event "3": {3}
      • From the event "factor of 54": {1, 2, 3, 6}
    • By combining, we have: \[ {1, 2, 3, 6} \]
    • Notice that 3 is already included in both sets.

  4. Unique outcomes: The unique outcomes for the event "3 or factor of 54" are: \[ {1, 2, 3, 6} \] This gives us a total of 4 unique successful outcomes.

  5. Calculate the probability:

    • The total number of outcomes when rolling a 6-sided die is 6.
    • The number of successful outcomes is 4.

Thus, the probability is given by: \[ P(3 \text{ or factor of } 54) = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}} = \frac{4}{6} \]

  1. Simplify the fraction: \[ \frac{4}{6} = \frac{2}{3} \]

Therefore, the simplified answer is: \[ \boxed{\frac{2}{3}} \]