Permutations and combinations are important concepts to understand in order to estimate probability and make better decisions. Behind the Seahawks fateful Superbowl decision are permutations and combinations that described the team’s different play call sequences.

 

We can assess the Seahawks play call decision to understand probability concepts. Let’s analyze what happened. In doing so, we will discuss two concepts underlying probability: combinations and permutations.  Facts: 1 timeout. 2nd down, which meant there were a maximum of 3 plays remaining. The play clock stops when there is an incomplete pass. The play clock keeps running on a run that does not result in a touchdown.

What Were the Play Call Possibilities

Combinations: The coaching staff had 4 possible play combinations. In statistics, we say that combinations are how many ways something can be arranged, where order doesn’t matter. Tossing a coin 3 times can result in 4 possible combinations: 3 heads, 3 tails, 2 tails and 1 head, or 1 head and 2 tails. Given the Seahawks had 3 plays remaining, the Seahawks coaching staff similarly had 4 combinations:

    1. 3 runs
    2. 3 passes
    3. 2 passes and a run
    4. 2 runs and a pass

Decisions: The first decision they had to make was which one of these combinations would give them the best chances. They knew that running the ball would give them the best chances. Therefore, two of the combinations that had more passes than runs could be eliminated, namely 3 passes, and 2 passes and a run. That left two possible combinations: 3 runs, and 2 runs and a pass. Pete Carroll has said that they knew they had to pass it once. This is because it was not possible to run the ball three times. If they ran the ball on the second down and it did not result in a touchdown, the clock would keep running, and they would have had to take their only timeout. If the second run on third down also did not make it in, the clock would keep running and they might have had one more chance on fourth down, but they were at the mercy of the clock. The three pass sequence did not guarantee they would have 3 downs. The sequence of 2 runs and 1 pass guaranteed that they would have 3 downs. That is the rational for Carroll’s “pass it once” comment.

Permutations

Permutations are the number or ways something can occur in a given order. It is often said that “order matters”.  Here are the permutations available to the Seahawks to choose from:

There is 1 permutation of 3 runs.

There is 1 permutation of 3 passes.

There are 3 permutations of 2 runs and 1 pass:

    1. Run-Pass-Run
    2. Pass-Run-Run
    3. Run-Run-Pass

There are 3 permutations of 2 passes and 1 run:

    1. Run-Pass-Pass
    2. Pass-Run-Pass
    3. Pass-Pass-Run

Thus, the Seahawks had 4 combinations and 8 permutations. The first decision was which combination to run. The combination of 2 runs and a pass gave them the best chances to run 3 plays and use the run as many times as possible.

Combinations Number of Permutations
3 Passes 1
3 Runs 1
2 Pass 1 Run 3
2 Runs 1 Pass 3
Total 8

We can see with 3 downs left, there were 8 permutations of plays they could potentially run.  This is similar to tossing a coin 3 times which also has 8 permutations:

1. H-H-H    2, T-T-T    3. H-T-T    4. T-H-T    5. T-T-H,    6. H-H-T     7. H-T-H     8. T-H-H

There is a simple equation to estimate the number of permutations:  n^r, where there are n things to choose r times.   2^3 = 8

Knowing the number of permutations helps us estimate probability.  When flipping a coin, there is an equal chance of landing on heads or tails.

  • Probability of 3 tails                                      1/8, or 12.5% probability
  • Probability of exactly 2 heads                       3/8, or 37.5% probability
  • Probability of heads, tails, heads                  1/8, or 12.5% probability

This is easy because heads and tails have equal probability. It would be a different analysis for the Seahawks because running and passing have different probabilities. The Seahawks had additional considerations in evaluating probability, based on time remaining, number of timeouts, and how the defense lined up to name a few.  But the first decision was which combination and permutation to choose.

I believe in order to the Seahawks to be guaranteed 3 plays (i.e. 3 chances to win), it was impossible to do 3 running plays. If run 1 did not get in, the clock would keep running. At best they would get a second run before time runs out and maybe, but only maybe, have time to use the 1 timeout remaining. Running 3 times was not possible given the time remaining and just 1 timeout.  Therefore, they had to pass the ball at least once. There were three ways to do that:

  • Run-Run-Pass
  • Run-Pass-Run
  • Pass-Run-Run

Run-Run-Pass: If run 1 didn’t get in, they would have had to hurry to get a second run and hope they could get it off with some time left to take a timeout have a final third play. This was highly uncertain. The more likely option is that they would have to take a timeout after second down, leaving them likely just one last play on third down. If they were successful in being able to get a third play off while the clock was ticking towards zero, they most likely would have run it again.

Run-Pass-Run: If the first run did not result in a touchdown, they would likely have had to take a timeout. If the pass did not result in a touchdown, the clock would stop. So this scenario would give them the option to regroup after each down and not do a “hurry up” offense. Then they would have one final attempt on fourth down. However, after the first run, it is almost 100% certain to the defense that a pass would be the next play because they would have to stop the clock. This sequence is the most likely and most predictable for the defense.

Pass-Run-Run: This gave them the best chances to guarantee 3 plays. The defense was not expecting this option, so it was a defensible play call give the other options.