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Monday, March 19, 2007

Howard versus Stoughton in 2007 Brier: The 9th End

With all due respect to Linda Moore and Ray Turnbull, the recent game of Howard versus Stoughton during the round robin game of the 2007 Tim Horton’s Brier showed how even some of the most knowledgeable don’t understand statistics as it relates to curling.

Howard is one up without hammer in 9th end. Two rocks remain. There is a corner guard and a Stoughton rock on the opposite side of the sheet in the back twelve foot. Richard Hart, the Ontario third, and Glen Howard discuss the options. Glen prefers a freeze to the back stone, but eventually Richard convinces him to play a draw around the corner guard. Rich’s argument is their uncertainty of the ice on the freeze attempt – Glen’s concern is the amount of curl he will get with his come-around attempt.

Linda’s comments were something to the effect “Howard wants Stoughton to score here and prefers giving up a deuce to a blank”. Perhaps Linda did not properly express what she meant or in fact, does not recognize the real numbers.

Howard would, mathematically, prefer a blank over Stoughton’s deuce. If, as the announcers stated, Howard preferred a deuce to a blank – why didn’t Stoughton take out his own rock rather than draw for two?

Recall the statistics:

Expected Results in the 10th end
Odds of winning if tied with hammer (x) = 75.7%
Odds of winning if one down with hammer (y) = 39.5%
Odds of winning if two down with hammer (z) = 11.7%

If Howard is one up, he should win about 60% and if he’s one down with hammer he will win 40%. It is clear that a blank is preferred to surrendering a deuce.

Then why, might you ask, does he not hit the Stoughton stone out?

What the TSN crew perhaps meant to say, or misunderstood, is:

Howard is willing to risk a deuce by Manitoba because the chance that he may either steal or force Stoughton to one increases his chance to win.

If Howard steals, he wins 88% of the time. If Stoughton is forced to one, he wins 75%. The chance that one of these outcomes may happen, overcomes the difference between 60 and 40 for one down versus one up.

Let’s examine the Ontario decision and also determine if Richard or Glen had the correct call.

From my previous article, “One up in Nine without Hammer” if we expect to give up a deuce roughly less than half the time, we should draw around the guard. I assume we all agree Glen can be expected to make a successful draw (where Stoughton cannot score two) more often than 50%. I’d suspect it’s as high as 80%, but he was unsure of the ice and considered the freeze instead. Because a steal is less likely (corner guard instead of centre guard) – it’s likely higher, say even 60% (see previous article for more precise numbers).

Referring to the article “Aggressive Play in 9th end”, where no guard was available, I roughly determine that Howard needs to make either a freeze or “near-freeze” and avoid a deuce at least 80% of the time.

Based on these numbers, our rough estimate to attempt a come-around is the correct shot.

Let’s examine a little closer…

Assuming Stoughton will blank 95% of the time Howard attempts a hit, we know from previous articles:

W = 61%

Assume a steal will not occur.
a = Stoughton takes one point = 75%
b = Stoughton scores two points = 25%
W = ax + by
= 67%

Assume a steal will not occur.
a = Stoughton takes one point = 50%
b = Stoughton scores two points = 25%
c = Stoughton blanks = 25%
W = ax + by + c(1-y)
= 63%

This shows us mathematically that Richard’s call is correct. If you include the slight chance at a steal, we could also expect it more likely with the come-around than the freeze.

A few observations:
Blank is not likely with the come-around. A freeze attempt makes a blank very possible.

A deuce is, perhaps in Glen’s opinion, more likely to occur with a missed come-around. Though I’ve used 25% in both calculations for a deuce, Howard may determine he is more likely to give up the deuce on a come-around and would prefer the freeze in order to add the possibility of a blank.

Ice conditions – Glen was not comfortable with the ice and with the shot and that can have an impact (though not mathematical). In hindsight, appears to be the case, Glen put his draw through the rings.

Assuming acceptable ice conditions, I would play the come around every time. What would you do?

Why do some players and possibly announcers believe it is preferable to be one down with hammer?

Statistics show that one up without wins roughly 60% of the time. Why would skips choose to be one down with hammer instead? I have two theories:

They don’t know it’s 60%…and more importantly

They want the last rock.

Top skips believe last rock puts the outcome in their hands. What they fail to realize, I suspect, is that very often the end results in no opportunity at two and, sometimes, a very difficult shot for one in order to force an extra end. In the example above, Howard needed to make a double on his last shot to score one and had no realistic chance at two as the end developed.

My belief is the preferred position is actually one up without hammer in the 9th end.

Four rock free-guard zone allows the team without hammer to force the play. The risk of a three exists, but I suspect the opportunity to steal or force your opponent to a single, and greatly increase your chance to win, is much higher. You are able to aggressively play the end with the risk of giving up a deuce and still having a 40% chance to win in 10.

We have not yet derived the statistics to analyse 9th end results, but once we do I will be examining the numbers to determine if they support or dispute my theory.