"What are the odds?" Bar Games Edition

Received two bar games-related questions in the past few days - both fitting "what are the odds?" style probability inquiries.

Yahtzee $5 Dice Roll

First: the game is a dice roll, with the aim to roll 5 dice to get a Yahtzee (all 5 dice match). You get 5 attempts, but unlike Yahtzee, you don't get to keep any matching dice - you have to reset your roll each time. Entry fee is $5, and the winner gets the pot of all entry fees (minimum pot size is $50). What does the pot size have to be for you to want to enter?

What we're looking for is at what point the expected value (E[V]) is positive. The odds of rolling a Yahtzee on one roll is if any one number matches on 5 dice, with 6 possible number matches. Thus (1/6) ^ 5 = 1 / 7776, times 6 = 6 / 7776 = 1 / 1296, or 0.08%. Verified here. Thus the odds of NOT winning on one roll is 1 - 0.08% = 99.92%.

Since there is no rollover, the math is pretty straight forward. The odds you LOSE 5 times is 99.92% ^ 5 = 99.6%. So the odds you WIN is 1 - 99.6% = 0.4%.

Your expected loss is 99.6% * -$5 = -$4.98, so we want to calculate at what point your expected winnings would be more than $4.98, and thus create a breakeven point. This can be calculated by your expected loss / odds you win, so $4.98 / 0.4% = $1,293. This is how large the pot needs to be before you'd be indifferent between lighting $5 on fire and playing the dice game; way higher than the $50 minimum pot size.

Odds You Miss Every Single Trivia Question

Second: the game is bar trivia, with 20 questions with 4 multiple choice answers each. Say you're trying to tank the game and miss every single question (for draft position next round maybe??) and you christmas-tree each question at random. What are the odds you get at least one question correct by accident?

If each guess is truly random, you have a 75% chance of getting a single question wrong. So the odds you get EVERY question wrong are 75% ^ 20 = 0.3%, meaning that it is extremely likely you will get at least one question right, at 1 - 0.3% = 99.7%.

There is actually a better chance you win the dice game than miss every trivia question!

Cost Benefit of SodaStream Over Canned Sparkling Water

Note: This is NOT paid content for SodaStream... or Spindrift... or Sparkling Ice...

First we determined Lime is the best Spindrift flavor. Then we determined Black Cherry is the best Sparkling Ice flavor.

Now I'm going to illustrate why you should purchase none of these things, and get a SodaStream instead.

At the request of avid sparking water drinkers, I've been tasked with comparing the variable per unit cost of a drink from a SodaStream machine, vs purchasing canned sparkling water in bulk. So I tracked every one of my uses of a brand new SodaStream CO2 canister, until it went flat.

I tracked the number of drinks made, not the number of drinks actually consumed; just like if a can is opened and not finished, that can is still used.

Over 4 months, I got 63 uses out of 1 CO2 canister, which yielded 162 drinks. I can buy each replacement canister for $15 each. Since we're only considering the marginal variable cost, I'm treating the initial SodaStream cost as a fixed sunk cost.

If you want any flavor, we also have to consider something like Bubly drops (~3 bottles of drops would last an entire CO2 canister, which adds $15.33). Or, super bourgeoise, a fresh lime for each use - with a median of 3 drinks per SodaStream use, that's 54 limes at $0.50 each = +$27.

Our alternatives range from Costco brand sparkling water (Kirkland), LaCroix, Sparkling Ice, to Spindrift.

All fall to the SodaStream! Even when using fresh produce each time:

Brand	TotalCost	CostPerDrink
SodaStream	$14.99	$0.09
SodaStream w/ Drops	$30.32	$0.19
SodaStream w/ Limes	$41.99	$0.26
Kirkland	$9.83	$0.28
LaCroix	$10.99	$0.46
Spindrift	$17.04	$0.57
Sparkling Ice	$9.98	$0.83

Some more descriptive statistics:

Avg Pumps	Avg Drinks	PumpPerDrink	Uses	# Drinks
4.16	2.57	1.62	63	162

The Definitive Sparkling Ice Rankings

Note: This is NOT paid content for Sparkling Ice

Previously, I wrote a post definitively determining the number 1 Spindrift flavor to be Lime. I have this same data available for Sparkling Ice flavors, which can be found at places like Wal Mart and Publix, and aim to calculate an explicitly scientific rating like I did for Spindrift.

Compiling the ratings of 4 different people, I've calculated a robust, indisputable ranking of all flavors.

Everyone was asked to rate each flavor on a scale of 1-10, with 10 being the highest, and 5 being the average, and the average ratings result in the following expansive ranking set, including Classic flavors, the Caffeine flavors, and even limited edition flavors:

OrdinalRank	CardinalRank	Item	Brand	Notes
1	9.1	Triple Citrus	Ice Caffiene
1	9.1	Black Cherry	Ice
3	8.3	Lemonade	Ice
4	8.1	Strawberry Lemonade	Ice
4	8.1	Cherry Limeade	Ice
6	8.0	Blue Raspberry	Ice Caffiene
7	7.9	Lemon Lime	Ice
8	7.6	Black Raspberry	Ice
9	7.1	Pink Grapefruit	Ice
10	7.0	Black Raspberry	Ice Caffiene
11	6.9	Pomegrante Blueberry	Ice
12	6.8	Kiwi Strawberry	Ice
13	6.6	Strawberry Citrus	Ice Caffiene
13	6.6	Mandarin	Ice
15	6.5	Coconut Pineapple	Ice
16	5.6	Orange Passion Fruit	Ice Caffiene
17	5.5	Fruit Punch	Ice
18	5.3	Cranberry Frost	Ice	Limited Edition
19	5.1	Grape Raspberry	Ice
20	4.0	Coconut Limeade	Ice
21	3.8	Ginger Lime	Ice
22	3.3	Orange Mango	Ice
23	3.1	Peach Nectarine	Ice
24	3.0	Mystery - Passion	Ice	Limited Edition
25	1.0	Crisp Apple	Ice

Once again though, we need to go a step further - human bias definitely exists in this cardinal rating. My ratings don't have a mean, median, OR mode of 5. And with the median higher than the mean, my ratings are very negatively-skewed:

*Scale relative to all ICES, out of 10	Count	Mean	Median	Mode
MS	25	6.22	6.5	8

As I did with Spindrift to adjust for this, former guest poster Jason Laso used the technique of calculating each individual's percentile on each rating, which normalizes each person's rating set to be from 0 to 10 and centered around 5.

Doing this on all ratings provides a better assessment over all preferences, and gives a definitive ranking of:

OrdinalRank	AdjustRating	Item	Brand	Notes
1	9.5	Triple Citrus	Ice Caffiene
1	9.5	Black Cherry	Ice
3	7.9	Strawberry Lemonade	Ice
4	7.8	Cherry Limeade	Ice
5	7.6	Lemonade	Ice
6	7.6	Blue Raspberry	Ice Caffiene
7	7.3	Lemon Lime	Ice
8	6.9	Black Raspberry	Ice
9	5.9	Pink Grapefruit	Ice
10	5.8	Black Raspberry	Ice Caffiene
11	5.5	Pomegrante Blueberry	Ice
12	5.3	Kiwi Strawberry	Ice
13	5.1	Coconut Pineapple	Ice
14	5.0	Mandarin	Ice
15	4.7	Strawberry Citrus	Ice Caffiene
16	3.0	Orange Passion Fruit	Ice Caffiene
17	2.8	Fruit Punch	Ice
17	2.8	Grape Raspberry	Ice
19	2.6	Cranberry Frost	Ice	Limited Edition
20	1.8	Coconut Limeade	Ice
21	1.4	Ginger Lime	Ice
22	1.3	Orange Mango	Ice
23	1.0	Peach Nectarine	Ice
24	0.8	Mystery - Passion	Ice	Limited Edition
25	0.0	Crisp Apple	Ice

There's a tie at the top! Triple Citrus (Caffeine) and Black Cherry. Since it's one of the OGs, we'll give the tiebreaker to Black Cherry, with the exact same 4 scores - 10, 9.6, 9.6, and 8.8.

However, if we look at the data further, we come to a far more important, concrete conclusion - Crisp Apple is definitively the worst flavor ever created. A perfect straight 0 score - every single entrant's worst flavor.

Congrats to Crisp Apple, a flavor that doesn't even appear on their web site any more!!

"What are the odds?" The Chances of a Canadian Team Winning the Stanley Cup Are Helped By 2021's Unique Playoff Structure

This season, due to the ongoing pandemic, the 2021 NHL division alignment's were changed, such that the 7 Canada teams only played eachother, including over the first 2 rounds of the Stanley Cup playoffs. Notably, no Canadian team has won the Cup since 1993 - a 26 season drought. 

This year's playoffs guarantee a team from Canada will make the semifinals - which of course increases the chance of that team making the Finals (something that's only happened 5 times since 1993), and subsequently winning it all. So how does this help (or hurt) the chances of a Canadian team finally winning the Cup again?

I'm looking to isolate the impact of the postseason format, so it's not as simple as proportionally taking the number of Canadian teams over the number of teams in the league each year (for example, when the league added 2 American teams in 1993-93, 8 Canada teams / 26 total teams = 31% Cup pre-playoffs). Since Canada teams have not been evenly spread across divisions (such as none in the Metropolitan), I want to start the analysis with the initial playoff field each year.

Just accounting for the number of Canadian playoff teams, the best chances since '93 have been 1996, 2004, 2015, and 2017, when 5 teams made the playoffs. Proportionally that would be a 31% chance - 5 / 16. Using this methodology, there is only a 0.23% chance that no Canadian team has won in the past 26 playoffs.

This divisional discrepancy affects the playoff matchups too - so I also broke down each round by seeding. The team with home ice and the higher seed is expected to win about 64% of the time, and calculating this across the entire playoffs identifies 2017 as the single best chance for a Canadian victor - 34%  that year. Using this methodology, there is only a 0.47% chance of no Canadian team winning it all since 1993.

As mentioned before, the impact this year comes from the guarantee of a team in the semifinals. So I took this a step further and looked at the odds of having no Canadian teams in the semifinals.

This ranges from 2016, when no Canadian teams made the playoffs, to 2017, when there was an 80% chance of having >= 1 Canadian team in the semis. More than one team can make it that far too, but this doesn't outweigh the fact that in no year is there a 100% chance of 1 team. If every semifinalist has a 25% of winning from there, this only shrinks the odds further (full table below). Using this methodology, there is a 2.06% chance of no Canadian team finishing their run as champions in the past 26 seasons.

Season	Count CAD	Odds 0 in Semis	Odds >= 1 in Semis	Odds Win
1994	4	31%	69%	17%
1995	4	31%	69%	17%
1996	5	34%	66%	17%
1997	3	61%	39%	10%
1998	3	61%	39%	10%
1999	3	36%	64%	16%
2000	3	47%	53%	13%
2001	4	40%	60%	15%
2002	4	40%	60%	15%
2003	4	31%	69%	17%
2004	5	26%	74%	19%
2006	4	31%	69%	17%
2007	3	36%	64%	16%
2008	3	47%	53%	13%
2009	3	47%	53%	13%
2010	3	47%	53%	13%
2011	2	55%	45%	11%
2012	2	55%	45%	11%
2013	4	31%	69%	17%
2014	1	85%	15%	4%
2015	5	26%	74%	19%
2016	0	100%	0%	0%
2017	5	20%	80%	20%
2018	2	55%	45%	11%
2019	3	36%	64%	16%
2020	3	61%	39%	10%

Across all three methodologies, there are not many occurrences of Canada having a better than 1 in 4 chance to win it all. Even when considering the outcomes where more than 1 Canada teams reach the semifinals, the downside chance of no Canada team making it that far greatly reduced their odds in any given year. In this context, having at least a 1 in 4 shot at winning the Cup this year represents one of the best opportunities in almost 3 decades.

Season	Count CAD	Proportion Odds	Home/Away Odds	Home/Away, Semis-On Odds
1994	4	25%	25%	17%
1995	4	25%	25%	17%
1996	5	31%	24%	17%
1997	3	19%	11%	10%
1998	3	19%	11%	10%
1999	3	19%	21%	16%
2000	3	19%	16%	13%
2001	4	25%	20%	15%
2002	4	25%	20%	15%
2003	4	25%	25%	17%
2004	5	31%	29%	19%
2006	4	25%	25%	17%
2007	3	19%	21%	16%
2008	3	19%	16%	13%
2009	3	19%	16%	13%
2010	3	19%	16%	13%
2011	2	13%	13%	11%
2012	2	13%	13%	11%
2013	4	25%	25%	17%
2014	1	6%	4%	4%
2015	5	31%	29%	19%
2016	0	0%	0%	0%
2017	5	31%	34%	20%
2018	2	13%	13%	11%
2019	3	19%	21%	16%
2020	3	19%	11%	10%