I’m curious what the chances are.
The flip comes down 8-9-10, and the player calls. Note that he does not mention whose suits he is talking about. All he remembers about the hand is that it included two jacks and a queen. The total amount of possibilities that an 8, 9, and 10 may occur on the flop is 4*4*4 = 64 combinations. 50 cards remain in the deck that are not known. Combin(50,3) = 19,600 is the number of possible ways to choose three cards from a deck of 50 cards. This means the chance of correctly calling the flip is 64/19,600 = 1 in 306.25.
On the turn, there are 47 unknown cards left in the deck, which means that the game is over. One in every 47 chances of getting it right were in his favor when he accurately predicted the four of spades.
There are 46 unknown cards remaining in the deck after the flop on the river. A 1 in 46 probability of getting it right was taken away when he properly predicted the two of hearts.
A 1 in 662,113 chance of getting all three predictions correct was (46/19600) (1/47) (1/46) = 1 in 662,113 chance of having all three predictions correct.
I’ll tell you what I believe. Unfortunately, it was just bad timing. At the gaming tables, I’ve witnessed a lot of people make erroneous predictions. Consider the millions of erroneous forecasts that were soon forgotten. Do you believe that anybody has created a YouTube video of these mistakes? Or, to put it another way, I’m disappointed.
On any regular season day in the National Hockey League, when a game is completed in normal time, the winner receives two points and the loser receives zero points. When a game goes into overtime, the winner will still get two points, while the loser will only receive one point for their efforts. The playoffs, on the other hand, provide no such incentive to force a game into extra time.
Consider the following scenario: A game is tied late in the game, during the regular season, do you believe that both teams would attempt to kill the clock in order to force the game into overtime? Due to the fact that three points would be given between the two teams instead of two, it would seem to be reasonable to proceed in this manner.
Indeed, for the reasons you mention, there seems to be an incentive in hockey to force a game into OT. For the sake of responding to your query, let’s have a look at various data sources. From the 2017/2018 season forward, the statistics shown here are derived from four seasons of ice hockey play.
The 7,846 games played over the course of four seasons are broken down into categories based on whether they were regular season or playoff games, and whether or not they went to extra time. According to the chart, 11.27 percent of games went into overtime during the regular season, while 54/544 games (or 9.03 percent) went into OT during the playoffs.
If the difference between 11.27 percent and 9.03 percent is statistically significant, or if it can be explained by normal variation, the answer is “yes.” Using a chi-squared test, I’ll compare the means of two different sample groups, much as you can do with the Comparison of proportions calculator on MedCalc.org. With 871 games going into overtime, there was an 11.10 percent chance that the game would go into overtime. According to the same sample, the likelihood of not working overtime is 88.90 percent.
In the unlikely event that there is no statistically significant difference between regular season and postseason games, then 804.6 regular season games and 66.4 postseason games should have gone to overtime, respectively.
After assuming that the real likelihood of overtime occurs at the same rate in both regular season and postseason games, the following table compares the actual outcomes to those predicted. The chi-squared statistic, which is defined as the square of the difference between the actual and anticipated totals divided by the expected total, is shown in the right-hand column of this table.
The chi-squared statistic of 2.813628 is shown in the table above. With just one degree of freedom, the chance of getting outcomes that are as skewed or worse is 9.347 percent. Therefore, assuming there is no change in behavior between a regular season and postseason game, resulting in genuinely equal chances of going into overtime, the likelihood of seeing a 2.24 percent difference in games going into overtime or greater is 9.347 percent. Alternatively, to put it another way, the data suggests that there is a statistically significant difference in the frequencies of overtime in the two types of games. However, there is still a 9.35 percent probability that it may be explained by normal random variation in the data set.
Please note that the chi-squared statistic is adjusted by a “N-1” factor in the MedCalc calculator that I linked to, as well as in other sources. It is specifically multiplied by (N-1)/N, where N is the total number of observations, and the resulting statistic is called the chi-squared statistic When applied to this situation, the modified chi-squared statistic is 2.813628 * (7845/7846) = 2.813270. 9.349 percent of the population is represented by this chi-squared statistic with a single degree of freedom. Because of this small change, I’m sure my readers would be perplexed as to why I didn’t make the change sooner rather than later.
For my part, I think that teams play more aggressively to force overtime in the regular season than they do in the playoffs, and the statistics lend credence to this belief, but the evidence does not prove it beyond any reasonable question.