NBA playoff matches are progressing these days, and it is time we look at the winning probabilities. How significant is the role of chance in these games? Are the outcomes independent and random with equal probabilities for wins and losses, like coin-tossing?
Equal probability scenario
Although the title speaks about randomness, what we examine here is equal probability outcomes. We will formulate hypotheses, calculate expected probabilities and test them using historical data. Playoff matches take the best-of-seven format. In this structure, the team that reaches four wins first takes the round (series) and goes to the next.
Scenario 1: 4 games
There are only two ways the round can end in four games: if A wins all four or if B does the same. P(AAAA) = (1/2)4 = 0.0625 (if the games are independent). P(BBBB) = (1/2)4 = 0.0625. The probability that the game ends in four games is 0.0625 + 0.0625 = 0.125.
Scenario 2: 5 games
The possibilities of A winning in five games are BAAAA, ABAAA, AABAA and AAABA. The last option, AAAAB, doesn’t exist as team A got the necessary four before reaching the potential fifth. The probabilities of each of these outcomes are (1/2)5, and there are eight possible outcomes (4 each for A and B). The probability that the game ends in four games is 4x(1/2)5 + 4x(1/2)5 = 0.25.
Scenario 3: 6 games
6-game winning options for A are BBAAAA, BABAAA, BAABAA, BAAABA, ABBAAA, ABABAA, ABAABA, AABBAA, AABABA, AAABBA. As before, the sequences AAABAB, AAAABB etc. are irrelevant. The probability that the game ends in six games is 10x(1/2)6 + 10x(1/2)6 = 0.3125.
Scenario 3: 7 games
Estimating the probability of a seven-game series is easy. You will get it by subtracting all the previous ones from 1 = 1 – (0.125 + 0.25 + 0.3125) = 0.3125.
Chi2 to the rescue
The null hypothesis, H0: The outcomes happen with equal probabilities, irrespective of the rank in the regular season. In other words, it is anybody’s game on a given day. The alternative hypothesis, H1, is that the game outcomes are not due to luck but based on the level of the team. We will do chi-squared statistics to test. That is next.
