there are four blue marbles, an unknown number of red (r) marbles, and six yellow marbles in a bag. which…

there are four blue marbles, an unknown number of red (r) marbles, and six yellow marbles in a bag. which expression represents the probability of randomly selecting a blue marble, replacing it, and then randomly selecting a red marble?\n○ $\frac{4}{10}(\frac{r}{10})$ \n○ $\frac{4}{10r}(\frac{r}{10r})$ \n○ $\frac{4}{10 + r}(\frac{r}{10 + r})$ \n○ $\frac{4}{10r}+(\frac{r}{10r})$

there are four blue marbles, an unknown number of red (r) marbles, and six yellow marbles in a bag. which expression represents the probability of randomly selecting a blue marble, replacing it, and then randomly selecting a red marble?\n○ $\frac{4}{10}(\frac{r}{10})$ \n○ $\frac{4}{10r}(\frac{r}{10r})$ \n○ $\frac{4}{10 + r}(\frac{r}{10 + r})$ \n○ $\frac{4}{10r}+(\frac{r}{10r})$

Answer

Explanation:

Step1: Calculate total number of marbles

The total number of marbles is the sum of blue, red and yellow marbles. There are 4 blue marbles, $r$ red marbles and 6 yellow marbles. So the total number of marbles is $4 + r+6=10 + r$.

Step2: Calculate probability of selecting a blue marble

The probability of selecting a blue marble on the first - draw is the number of blue marbles divided by the total number of marbles. So $P(\text{blue})=\frac{4}{10 + r}$.

Step3: Calculate probability of selecting a red marble

Since the marble is replaced, the total number of marbles remains the same for the second draw. The probability of selecting a red marble on the second - draw is the number of red marbles divided by the total number of marbles. So $P(\text{red})=\frac{r}{10 + r}$.

Step4: Calculate the combined probability

Since the two events are independent (because of replacement), the probability of both events occurring is the product of their individual probabilities. So $P(\text{blue then red})=\frac{4}{10 + r}\times\frac{r}{10 + r}$.

Answer:

C. $\frac{4}{10 + r}\left(\frac{r}{10 + r}\right)$