Here are the answers:

I assumed that each horse had a 20% chance of winning, based on a dividend of $5. The spreadsheet didn't automatically round bets to whole numbers, but the figures below are fairly accurate.

$5 target = need a win within 23 bets. 0.6% chance of going bust or once every 169 days. So a bust is over due!

$10 target = need a win within 20 bets. 1.15% chance of going bust or once every 87 days.

$20 target = 18 bets. 1.8% chance of going bust or once every 55 days

$30 target = 16 bets. 2.8% chance of going bust or once every 36 days.

$50 target = 14 bets. 4.4% chance of going bust or once every 23 days.