Question

a bag contains eleven equally sized marbles, which are numbered.
what is the probability that a marble chosen at random is shaded or is labeled with a multiple of 3?

\\(\bigcirc \frac{2}{11}\\)
\\(\bigcirc \frac{3}{11}\\)
\\(\bigcirc \frac{5}{11}\\)
\\(\bigcirc \frac{6}{11}\\)

Explanation:

Identify the sample space

Using the Sample Space knowledge point

$$ S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} $$

$$ n(S) = 11 $$

Identify the shaded marbles

Using the Favorable Outcomes knowledge point

$$ \text{Shaded} = \{1, 3, 4, 5, 9\} $$

$$ n(\text{Shaded}) = 5 $$

Identify the multiples of 3

Using the Favorable Outcomes knowledge point

$$ \text{Multiples of 3} = \{3, 6, 9\} $$

Find the union of both events

Using the Probability Addition Rule knowledge point

$$ \text{Shaded} \cup \text{Multiples of 3} = \{1, 3, 4, 5, 6, 9\} $$

$$ n(\text{Shaded} \cup \text{Multiples of 3}) = 6 $$

Calculate the probability

Using the Theoretical Probability knowledge point

$$ P(\text{Shaded or Multiple of 3}) = \frac{6}{11} $$

Answer:

• (A) \(\frac{2}{11}\)
• (B) \(\frac{3}{11}\)
• (C) \(\frac{5}{11}\)
• (D) \(\frac{6}{11}\) (Correct answer)