# Probability same football square wins all four quarters

A local casino has a Super Bowl football squares promotion going on. All the 100 squares are guaranteed to be filled by the customers. The casino will pay a certain amount to the winner of each quarter and then is thinking of paying a large bonus if the same square wins all four quarters. I started thinking about estimating the probability that the same square will win all four quarters. See if you agree witH my analysis as follows: The first quarter score is guaranteed to be won since all 100 squares are filled. And that same first quarter square will win the second, third, and fourth quarters if and only if these other three quarters have a score of "0"(e.g., 0, 10, 20, 30 etc) for each of the two teams. Now, the probability a teams score in any given quarter will be "0" can be estimated by anamlyzing previous games. In all 49 previous super bowl games there have been 49x4x2 = 392 quarterly scores by the teams involved. The number "0" was scored in 105 of these 392 quarterly scores. Thus the (approx) prob that a team scores "0" points in a quarter is 105/392 = 0.2679. So, the approximate prob that quarters Two, three, and four will result in scores of "0" for each team for each quarter is 0.2679x0.2679x0.2679x0.2679x0.2679x0.2679 = .0004 . Thus the prob that the same square will win all four quarters is approx .0004

## migrated from mathoverflow.netJan 12 '16 at 0:37

This question came from our site for professional mathematicians.

• Much about the situation you're describing is unclear -- indeed the second half I am not sure I followed at all. But even in the first half of the question it's not quite clear how the winning squares are obtained. (Are they equally likely to be winning squares, for example?) ... Could you edit your question to clarify how the thing works? Assume many of us don't know what these things are or how they work (I sure don't). – Glen_b Jan 12 '16 at 0:44
• I agree with your analysis Burt - pretty unlikely!!! – MikeP Jan 12 '16 at 16:19

The analysis is essentially correct but comes up short of being correct because 1) you also need to include an approximately 70% factor that the teams have a different first quarter score and, more importantly, 2) teams' scores from quarter to quarter are not independent. That is, consider a team that scores 0 in a quarter ... it is more likely to score 0 in a future quarter (the offense is not playing well). A quick count in the first 50 superbowls shows 82 times a team has scored at least one "0" (i.e. 0, 10, 20,...) in a quarter. 51 times of that 82 (0.622), the team has scored at least one other "0" in a quarter in the game.

So, a better estimate would be: 0.7 (first quarter anything but a tie) x 0.2679^2 (both teams score a 0 in Q2) x 0.622^4 (both teams score "0"s in Q3/Q4 given they already have a quarter equal to 0) = 0.75%. This is much larger than your 0.04%.

Actually, 4953 professional football scores have been analyzed (footballsquares.blogspot.com) and used to simulate all the score correlations that are present. The simulation finds 0.24% of a probability of the same square winning all four quarters. Indeed 12 out of 4953 (very close to 0.24%) games had a single box win all four quarters.