Last week an interesting thread popped up on the Rokslide forum titled "Do you really understand Nevada Draw Odds?" in which Gohunt's Trail Kreitzer gave an explanation of the math behind how Gohunt calculates some of their numbers. Very early in the thread the question arose about how the order of your 5 hunt choices might affect the draw odds of each individual choice. That is something we've looked at extensively here at Toprut, and all of the data very strongly supports that you can reasonably estimate that with some pretty straight forward math. We figured we'd contribute to the conversation since it was highly relevant to the discussion and not something that is widely known.
Given that Gohunt is a sponsor on Rokslide, we private messaged the forum thread's moderator, Robby Denning, asking for permission to post the comment that we did. He agreed and said it was fine and we commented in post #19 of the thread. A few hours after we posted that to the thread, things started to get interesting.
Gohunt then responded and dismissed our entire post pointing out that sometimes the math as we described would produce a negative probability. They promptly threw the baby out with the bath water and also discredited our claim about how to figure your odds for the application as whole.
More independent confirmation about the simple math method to estimate your draw odds beyond choice 1, and for your application as a whole, can be found in the Arizona Game & Fish department's publication, "Hunt Arizona, 2016 Edition". See page 1 under the section Beating the Odds (Arizona'a draw also considers multiple choices before moving on to the next application).
From there the thread predictably turned a few different directions with some lingering questions amongst some and what appeared to be misinterpration from others. Things appeared to be coming to a conclusion. And then Trail added some details of how Gohunt figures draw odds in post #44 of the thread. And this is where things got really interesting for us. We've known for a few weeks that Gohunt's Nevada draw odds have a big problem (New Mexico too), and now we have some insight into how they are looking at things.
- Before the draw begins, each app is assigned a "sequence" number, based on randoms.
- The number of random numbers you get is determined by your bonus points squared + 1.
- Your application is assigned the lowest of those random numbers. This is your sequence number. All others are discarded.
- All apps are sorted based on the sequence number, and then applications are evaluated in order, from smallest sequence number to largest.
- Your chances of getting drawn are ultimately completely dependent on the sequence number assigned to your application.
- Each of your hunt choices are not treated as an "application". Every application is pulled and evaluated exactly one time.
- When your application is pulled, they look at all your choices, in order, before moving to the next application.
One of the main points that Gohunt made in the thread is that the chances of drawing choices 2 through 5, are affected in the same relative manner as if you flipped a coin 5 times trying to get a single result. "Compounding probabilities" they claim is how you calculate odds for choices 2 through 5 - just like a coin flip. There's just one huge problem: coin flips are independent events, and the chance that you draw a tag with hunt choices 2 through 5 on your Nevada application ARE NOT. They are dependent events - dependent on your application's sequence number and dependent on the order of your choices. We're not flipping coins here!
Compound Probabilty of Independent Events
Introduction to Dependent Probability
Dependent Probability, Example 1
The examples given in the forum thread are for a hypothetical application where you have 5 hunt choices, with 1st choice draw odds of 10, 15, 20, 25 and 30% respectively.
In Gohunt's explanation of how you should apply compounding probablilites to draw odds for the scenario given, they throw in some ridiculous marketing language and said "our data scientist ran the scenario toprut provided through 100 million simulations just to make sure we are confident and the results listed below are solid". We'll save you the 100 million simulations. If you have a calculator, here's how you get to their (incorrectly applied) results from post #44 of the thread:
1st choice odds = 0.1 -> ([10/100]) { the listed odds for this hunt }
2nd choice odds = 0.135 -> ([90/100] * [15/100]) { | the chance we WON'T draw all previous choices * the draw odds for this hunt }
3rd choice odds = 0.153 ---> ([90/100] * [85/100] * [20/100]) { | }
4th choice odds = 0.153 ---> ([90/100] * [85/100] * [80/100] * [25/100]) { | }
5th choice odds = 0.138 ---> ([90/100] * [85/100] * [80/100] * [75/100] * [30/100]) { | }
And in the other example when the choice order is reversed, easiest to hardest:
1st choice odds = 0.3 -> ([30/100]) { the listed odds for this hunt }
2nd choice odds = 0.175 -> ([70/100] * [25/100]) { | the chance we WON'T draw all previous choices * the draw odds for this hunt }
3rd choice odds = 0.105 ---> ([70/100] * [75/100] * [20/100]) { | }
4th choice odds = 0.063 ---> ([70/100] * [75/100] * [80/100] * [15/100]) { | }
5th choice odds = 0.036 ---> ([70/100] * [75/100] * [80/100] * [85/100] * [10/100]) { | }
Feel free to punch each of those results into a handheld calculator many millions of times just to make sure the results are always the same.
The easiest way to visualize that each choice is not an independent event (ie coin flip), is to consider the case where you have reversed your choice order and put the easiest to draw 30% hunt first, and the hardest to draw 10% hunt as your 5th choice. Let's say your sequence number is at the 40th percentile of all application sequence numbers. They pull your app, after already looking at 40% of all other applications, and determine that your 1st choice 30% odds hunt has no tags remaining to give out. Your random sequence number was just too high and you didn't get pulled early enough to get your first choice. And now your second choice is an even harder to draw tag. How could you realistically expect to draw that second choice, given that it is in even higher demand? The answer of course, is that you can't.
Gohunt concluded you have a 17.5% chance for that harder 2nd choice in the example given. In reality your chances are almost always 0%.
Let's look at some real facts to back that up.
We analyzed all of the 2016 antlered mule deer applications for both residents and nonresidents. We found 11,485 real instances where an application did not draw their easier 1st choice, and had specified a harder to draw 2nd choice. To determine easier/harder hunts, we mapped each individual applications bonus point level and hunt choices to the actual draw odds for each hunt. Of those 11,485 instances, ONLY 7 were successful in drawing that 2nd choice (that's 0.06%). Of those 7 that overcame the odds, the average difference between the listed draw odds of choice 1 and choice 2 was only 0.82%. In other words, the people that did actually draw a harder choice 2 after an easier choice 1 had hunt choices that were very close to having the same odds in the first place.
After reading Gohunt's misinterpration of how multi-choice draw probabilities should be calculated, we can only conclude that the "data scientists" have misunderstood some critical aspects of the Nevada big game draw on a more fundamental level. And that explains a lot - since we already know that Gohunt's Nevada draw odds are systematically and significantly too low.
