Question

if each marble is replaced after it is drawn, what is the probability of randomly drawing three consecutive red (r) marbles?
\\(\frac{3}{256}\\)
\\(\frac{1}{56}\\)
\\(\frac{1}{27}\\)
\\(\frac{27}{512}\\)

Explanation:

Step1: Count total marbles

There are 9 marbles in total.

Step2: Determine probability of drawing a red marble

There are 3 red marbles. The probability of drawing a red marble in one draw is $\frac{3}{9}=\frac{1}{3}$ since probability = $\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$.

Step3: Calculate probability of three - consecutive red draws

Since the draws are independent (because the marble is replaced after each draw), the probability of three consecutive red draws is the product of the probabilities of drawing a red marble in each individual draw. So $P=\frac{1}{3}\times\frac{1}{3}\times\frac{1}{3}=\frac{1}{27}$.

Answer:

$\frac{1}{27}$