You choose a marble at random from a bag containing 10 red marbles, 5 yellow marbles, and 1 pink marble. You replace the marble and then choose again. Find the probability of P(yellow then red)

To find the probability of selecting a yellow marble followed by a red marble, we multiply the probabilities of each event happening individually because they are independent events. The probability of selecting a yellow marble is \( \frac{5}{10 + 5 + 1} \) because there are 5 yellow marbles out of a total of 10 red, 5 yellow, and 1 pink, and we replace the marble afterward. The probability of selecting a red marble is also \( \frac{10}{10 + 5 + 1} \) because there are 10 red marbles out of the same total. Therefore, the probability of selecting a yellow marble followed by a red marble is: \[ P(\text{yellow then red}) = P(\text{yellow}) \times P(\text{red}) \] \[ = \left(\frac{5}{10 + 5 + 1}\right) \times \left(\frac{10}{10 + 5 + 1}\right) \] \[ = \left(\frac{5}{16}\right) \times \left(\frac{10}{16}\right) \] \[ = \frac{50}{256} \] \[ = \frac{25}{128} \] So, the probability of selecting a yellow marble followed by a red marble is \( \frac{25}{128} \).