Elizabeth brought a box of donuts to share. There are two-dozen (24) donuts in the box, all identical in size, shape, and color. Two are jelly-filled, 5 are lemon-filled, and 17 are custard-filled. You randomly select one donut, eat it, and select another donut. Find the probability of selecting a custard-illed donut followed by a lemon-illed donut. \( \square \) (Type an integer or a simplified fraction.)

To find the probability of selecting a custard-filled donut followed by a lemon-filled donut, we can break it down into two steps. First, the probability of selecting a custard-filled donut is the number of custard donuts (17) divided by the total number of donuts (24): \[ P(\text{Custard}) = \frac{17}{24} \] After eating the custard-filled donut, there are now 23 donuts left: 16 custard-filled, 5 lemon-filled, and 2 jelly-filled. The probability of then selecting a lemon-filled donut is: \[ P(\text{Lemon} | \text{Custard}) = \frac{5}{23} \] Now, to find the total probability of both events happening, we multiply the probabilities together: \[ P(\text{Custard and then Lemon}) = P(\text{Custard}) \times P(\text{Lemon} | \text{Custard}) = \frac{17}{24} \times \frac{5}{23} = \frac{85}{552} \] This fraction can be simplified to: \[ \frac{85}{552} = \frac{85 \div 17}{552 \div 17} = \frac{5}{32} \] So the probability of selecting a custard-filled donut followed by a lemon-filled donut is \(\frac{5}{32}\).