Dependent and Independent Events 1. A bag contains 3 red marbles and 5 blue marbles. You dram one marbe replace it, and then draw another. What is the probability that you dram a red marble both times?

To determine the probability of drawing a red marble **both times** when you replace the marble after each draw, follow these steps: 1. **Total Marbles in the Bag:** - **Red Marbles:** 3 - **Blue Marbles:** 5 - **Total:** \( 3 + 5 = 8 \) marbles 2. **Probability of Drawing a Red Marble on the First Draw:** \[ P(\text{First Red}) = \frac{\text{Number of Red Marbles}}{\text{Total Number of Marbles}} = \frac{3}{8} \] 3. **Since the Marble is Replaced:** - The total number of marbles remains **8** for the second draw. - The probability of drawing a red marble on the **second draw** is the same as the first: \[ P(\text{Second Red}) = \frac{3}{8} \] 4. **Probability of Both Events Happening (Independent Events):** - Since the marble is replaced, the two events are **independent**. Therefore, multiply the probabilities of each event: \[ P(\text{First Red and Second Red}) = P(\text{First Red}) \times P(\text{Second Red}) = \frac{3}{8} \times \frac{3}{8} = \frac{9}{64} \] **Final Answer:** \[ \boxed{\dfrac{9}{64}} \]