Chance

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Probability Calculations

Calculating Probabilities for a Single Season

The probabilities of winning a championship, appearing in a championship game or series, and making the playoffs are calculated under the assumption that all teams have equal ability. While factors such as teams’ schedules, if not balanced, could have a small impact on these probabilities, that impact is likely negligible. As a result, the calculations are based solely on the league structure in place at the time, including the number of teams, conference and division alignments, playoff qualification rules, and playoff format.

Playoffs

Consider a league in which division winners automatically qualify for the playoffs and additional teams qualify as wild cards. The first step is to determine each team's probability of earning an automatic playoff berth. This probability is simply:

Probability of winning the division = 1 ÷ number of teams in the division

In the NHL's current format, the top three teams in each division qualify automatically. In that case, the probability becomes:

Probability of automatic qualification = 3 ÷ number of teams in the division

Because divisions are not always the same size, teams may have different playoff probabilities.

The next step is to calculate the probability of qualifying as a wild card team. This probability depends on the size of the team's division, the number of teams in its conference or league, and the number of available wild card spots. For a more information related to the wild card calculations, click here.

Throughout the remainder of this section, the probability of making the playoffs will be denoted as P(PO).

Championship Appearances and Championships

In playoff formats without byes, calculating the probability of reaching the championship round is straightforward. Assuming all teams are equally skilled, each playoff series is effectively a 50-50 proposition.

Therefore, the probabilities are as follows for a given number of rounds:

2 playoff rounds, then P(CA) = P(PO) ÷ 2

3 playoff rounds, then P(CA) = P(PO) ÷ 4

4 playoff rounds, then P(CA) = P(PO) ÷ 8

where P(CA) represents the probability of making a championship appearance.

Once a team reaches the championship round, it has a 50% chance of winning the championship. Therefore:

Probability of winning a championship (P(CH)) = P(CA) ÷ 2

Playoff Formats with Byes

Leagues that award first-round byes require additional calculations.

Consider a four-round playoff structure in which some teams receive a bye directly into the second round. In this case, teams that receive a bye need to win only two playoff series to reach the championship round, while teams without a bye must win three.

To calculate P(CA), we separately determine:

The probability of making the playoffs and receiving a bye.
The probability of making the playoffs without receiving a bye.

The first probability is divided by 4, reflecting the need to win two series to reach the championship round. The second probability is divided by 8, reflecting the need to win three series. Adding these two results together produces the overall value of P(CA).

The complexity of calculating bye probabilities varies by league and depends on how byes are awarded and how the playoff structure is organized.

Example

In the 1977 NFL season, each conference had two divisions with five teams each and one division with four teams. In each conference, the three division winners plus one wild card team qualified for the playoffs. The playoffs consisted of three rounds with no byes.

For the 5 team divisions:

Probability of winning the division = 1 ÷ 5 = 0.20000

Probability of earning a wild card spot = 0.07447

P(PO) = 0.20000 + 0.07447 = 0.27447

P(CA) = P(PO) ÷ 4 = 0.06862

P(CH) = P(CA) ÷ 2 = 0.03431

For the 4 team divisions:

Probability of winning the division = 1 ÷ 4 = 0.25000

Probability of earning a wild card spot = 0.06382

P(PO) = 0.20000 + 0.07447 = 0.31382

P(CA) = P(PO) ÷ 4 = 0.07846

P(CH) = P(CA) ÷ 2 = 0.03923

The calculations for the probability of earning a wild card spot are not shown here, but are based on order statistics methodology.

As demonstrated in this example, the chances of making the playoffs are higher for the divisions with four teams and subsequently have higher chances of making a championship appearance and winning a championship.

Calculating Probabilities for Spans of Seasons

For a given team and each span of seasons within the time frame being evaluated, the probability of the number of events or more occurring and the probability of the number of events or less occurring is calculated. Let’s look at an example using the Los Angeles Dodgers over the time frame of 1951 to 1969 with the event being Winning a World Series.

Probability of the Number of World Series Wins or MORE Occurs in a Given Span of Seasons for the Los Angeles Dodgers
World Series	Start Season					End Season
World Series	Start Season	1951	1952	1953	1954	1955	1956	1957	1958	1959	1960	1961	1962	1963	1964	1965	1966	1967	1968	1969
	1951	1.00000	1.00000	1.00000	1.00000	0.27580	0.32107	0.36350	0.40328	0.10492	0.12590	0.14775	0.16578	0.04018	0.04738	0.00985	0.01211	0.01468	0.01756	0.02023
	1952		1.00000	1.00000	1.00000	0.22752	0.27580	0.32107	0.36350	0.08503	0.10492	0.12590	0.14338	0.03211	0.03857	0.00746	0.00937	0.01156	0.01406	0.01639
	1953			1.00000	1.00000	0.17603	0.22752	0.27580	0.32107	0.06647	0.08503	0.10492	0.12170	0.02498	0.03068	0.00549	0.00707	0.00891	0.01103	0.01305
	1954				1.00000	0.12109	0.17603	0.22752	0.27580	0.04949	0.06647	0.08503	0.10094	0.01881	0.02374	0.00391	0.00517	0.00669	0.00846	0.01018
* WS Win *	1955					0.06250	0.12109	0.17603	0.22752	0.03440	0.04949	0.06647	0.08132	0.01359	0.01776	0.00266	0.00366	0.00487	0.00632	0.00775
	1956						1.00000	1.00000	1.00000	0.22752	0.27580	0.32107	0.35501	0.07767	0.09315	0.01676	0.02139	0.02663	0.03249	0.03786
	1957							1.00000	1.00000	0.17603	0.22752	0.27580	0.31201	0.05975	0.07408	0.01192	0.01579	0.02027	0.02536	0.03011
	1958								1.00000	0.12109	0.17603	0.22752	0.26615	0.04355	0.05651	0.00801	0.01114	0.01486	0.01919	0.02330
* WS Win *	1959									0.06250	0.12109	0.17603	0.21722	0.02936	0.04071	0.00499	0.00741	0.01039	0.01397	0.01745
	1960										1.00000	1.00000	1.00000	0.20679	0.24645	0.03796	0.05027	0.06376	0.07826	0.09110
	1961											1.00000	1.00000	0.15391	0.19621	0.02474	0.03532	0.04728	0.06046	0.07233
	1962												1.00000	0.09750	0.14263	0.01402	0.02259	0.03277	0.04438	0.05510
* WS Win *	1963													0.05000	0.09750	0.00725	0.01402	0.02259	0.03277	0.04245
	1964														1.00000	0.09750	0.14263	0.18549	0.22622	0.25846
* WS Win *	1965															0.05000	0.09750	0.14263	0.18549	0.21943
	1966																1.00000	1.00000	1.00000	1.00000
	1967																	1.00000	1.00000	1.00000
	1968																		1.00000	1.00000
	1969																			1.00000

To illustrate how these probabilities are calculated, consider the period from 1958 through 1964 (highlighted in blue). During those seven seasons, the Dodgers won the World Series twice. Therefore, we need to determine the probability of a team winning the World Series at least two times within that span. To calculate this probability, we must account for every possible combination in which the Dodgers could have won exactly two championships during those seven seasons. For example, they could have won in 1958 and 1959, 1958 and 1960, 1958 and 1961, and so on. We must also include the probabilities of winning exactly three, four, five, six, or all seven championships during the span. Adding together the probabilities of all these outcomes produces the value shown in the table, 0.05651.

If the Dodgers had the same chance of winning the World Series every year, this would be a straightforward calculation using the binomial distribution. However, the yearly probabilities were not constant. As a reminder, for the purposes of this website, we are assuming all teams have equal ability. Therefore, from 1958 through 1961, the Dodgers' probability of winning the World Series was 0.0625. Beginning in 1962, four expansion teams were added to Major League Baseball, reducing the Dodgers' probability to 0.05 for the 1962–1964 seasons. While the actual calculations performed for this website use more efficient methods, those methods produce the same result as calculating every possible combination separately and then summing the probabilities.

The table above shows these probabilities for every possible time span between 1951 and 1969. As another example, consider the period from 1954 through 1967. During those fourteen seasons, the Dodgers won the World Series four times. In this case, we need to calculate the probability of winning the World Series at least four times during that span. The table shows this probability as 0.00669 (highlighted in yellow). The key point here is that the probability being calculated depends on the number of championships actually won during the selected period.

Probability of the Number of World Series Wins or LESS Occurs in a Given Span of Seasons for the Los Angeles Dodgers
World Series	Start Season					End Season
World Series	Start Season	1951	1952	1953	1954	1955	1956	1957	1958	1959	1960	1961	1962	1963	1964	1965	1966	1967	1968	1969
	1951	0.93750	0.87891	0.82397	0.77248	0.96560	0.95051	0.93353	0.91497	0.98459	0.97899	0.97244	0.96643	0.99383	0.99213	0.99869	0.99826	0.99774	0.99712	0.99651
	1952		0.93750	0.87891	0.82397	0.97847	0.96560	0.95051	0.93353	0.98923	0.98459	0.97899	0.97375	0.99556	0.99418	0.99910	0.99877	0.99836	0.99787	0.99737
	1953			0.93750	0.87891	0.98877	0.97847	0.96560	0.95051	0.99294	0.98923	0.98459	0.98011	0.99693	0.99583	0.99940	0.99916	0.99885	0.99846	0.99806
	1954				0.93750	0.99609	0.98877	0.97847	0.96560	0.99577	0.99294	0.98923	0.98552	0.99798	0.99714	0.99962	0.99945	0.99922	0.99892	0.99861
* WS Win *	1955					1.00000	0.99609	0.98877	0.97847	0.99778	0.99577	0.99294	0.98997	0.99875	0.99813	0.99978	0.99966	0.99949	0.99927	0.99904
	1956						0.93750	0.87891	0.82397	0.97847	0.96560	0.95051	0.93693	0.99068	0.98726	0.99828	0.99753	0.99658	0.99542	0.99426
	1957							0.93750	0.87891	0.98877	0.97847	0.96560	0.95353	0.99404	0.99135	0.99896	0.99841	0.99770	0.99680	0.99588
	1958								0.93750	0.99609	0.98877	0.97847	0.96817	0.99655	0.99454	0.99942	0.99905	0.99854	0.99787	0.99716
* WS Win *	1959									1.00000	0.99609	0.98877	0.98053	0.99827	0.99689	0.99972	0.99948	0.99914	0.99866	0.99813
	1960										0.93750	0.87891	0.83496	0.98247	0.97301	0.99721	0.99545	0.99316	0.99032	0.98746
	1961											0.93750	0.89063	0.99156	0.98429	0.99867	0.99750	0.99586	0.99370	0.99145
	1962												0.95000	0.99750	0.99275	0.99952	0.99884	0.99777	0.99624	0.99455
* WS Win *	1963													1.00000	0.99750	0.99988	0.99952	0.99884	0.99777	0.99650
	1964														0.95000	0.99750	0.99275	0.98598	0.97741	0.96892
* WS Win *	1965															1.00000	0.99750	0.99275	0.98598	0.97884
	1966																0.95000	0.90250	0.85738	0.82165
	1967																	0.95000	0.90250	0.86490
	1968																		0.95000	0.91042
	1969																			0.95833

The same process is used to calculate the probability of a team winning a certain number of championships or fewer during a given span of seasons. While the first table helps identify periods in which a team may have performed at an exceptionally high level, this second table highlights periods in which a team may have underperformed relative to expectations. In both cases, lower probabilities indicate outcomes that would be less likely to occur by chance alone. A very low probability suggests that the team's performance over that span was unusually strong or unusually weak rather than simply the result of good or bad fortune.

Although the examples above focus on World Series championships, the same methodology is also applied to World Series appearances (or appearances in the championship game or championship series for other sports) as well as playoff appearances.

Determining Best and Worst Spans

The best (number of events or more occurring) and worst (number of events or less occurring) spans are considered the spans of seasons which are the least likely to have occurred (ie. they have the lowest probability).

After making the calculations for all possible spans within the time frame (1951-1951, 1951-1952, …, 1951-1969, 1952-1952, 1952-1953, …, 1952-1969, …, 1968-1969, 1969-1969), we find the lowest probability to determine the best span. In this case, we can see that the Dodgers winning 4 times from 1955-1965 produced the lowest probability of 0.00266 (approximately 1 in 376) which is highlighted in green in the first table. The worst span for the given time frame is determined by finding the lowest probability from the second table. In this case, the worst span is 1951-1954 where the Dodgers won zero World Series where the probability was 0.77248, highlighted in red.

The Dodgers also won zero World Series in the span 1966-1969. However, due to there being more teams in the league in that span, the span of 1951-1954 had a lower probability of resulting in zero World Series. This illustrates the importance of accounting for the structure of the league.

For the best and worst span leaderboards, a team can appear multiple times. To avoid good or bad spans from having similar years that are involved, once the best/worst span for a team is determined, the next span under consideration must not overlap with any year from the previous span. For example, the Pittsburgh Penguins best span for making the playoffs was when they made 13 playoff appearances between 2007-2019. Their next best span was making 12 playoff appearances between 2008-2019, but it included part of the previous span, therefore it was not used. The next best span that did not overlap with 2007-2019, was 11 playoff appearances between 1991-2001. The top 5 playoff appearance spans for the Penguins are as follows:

Start Year	End Year	Playoff Appearances	Probability
2007	2019	13	0.00019
1991	2001	11	0.00366
1975	1977	3	0.21600
2021	2022	2	0.25000
1979	1982	4	0.32256