Let:
p = Number of Party Poopers.
w = Number of wins.
e = Party Poopers needed to end bonus.
Imagine the w wins in a row, like a loaf of bread. Then place the p Party Poopers like raisins equaly along the bread, creating the same distance between consecutive raisins as well as from each end. For example, if the bread was 12" long, and you had 3 raisins, you would place the raisins at the 3", 6", and 9" points, creating 4 segments of 3" each.
The expected number of wins picked is the product of the length of each segment and the number of Party Poopers needed to end the bonus, or e × (w/(p+1)). Let's look at an example.
Suppose there is a field of 40 icons with 8 Party Poopers. It takes three Party Poopers to end the bonus. That would leave 32 wins. So, p = 8, w = 32, and e = 3. The expected number of successful picks is 3 × (32/(8+1)) = 32/3 = 10.67.