You have to start with certain assumptions. You take as a given that the golfer is going to get a hole in one. Of course he is not likely to get it on a par 4 or 5 hole. However, I will assume that whether or not the third hole is a par three is randomly distributed. You simply need to answer the question as posed. What are the odds that all 3 of his holes in one would be on the same number hole. Obviously if you are playing with no holes in one in your life, the chance your first will be on a number 3 hole is 1/9 for a nine hole course, and 1/18 for an 18 hole course. So now your life history has one hole in one for every round you play in the future. As you start your round on a 9 hole course I ask you, "What is the chance that if you get a hole in one today, it will be on the same number hole as the previous ace you made?" The answer is 1/9. It is your lucky day and you do get an ace. The same calculation applies for all future rounds when going for your 3rd ace. You take the odds of each individual event expressed as a fraction or decimal and multiply them to get a result. Easier example.. What are the chances of rolling a single die and getting 6 three times in a row. For each individual roll it is 1/6. So the result is 1/6 times 1/6 times 1/6 or 1/216. It is a different question than asking what are the odds of rolling the same number three times in a row. Because the odds of rolling a number on the first roll is 6/6 because you are going to get some number, the question there changes to what is the chance of matching your second roll to the first, which is 1/6. So to say it another way, "the odds of rolling doubles is 1/6." And of course the odds of rolling three dice at once all having the same number is 1/36. But the odds of rolling 3 dice and getting all 6's is 1/216. And yes I have time on my hands.
If you want to factor in the odds of making a hole in one at all, you can multiple the previous result by the odds of a hole in one on a round. You will need to know how many par 3's are in the average course if you want to know the odds of an ace in a round to do the correction for a course with excess par 3's.
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