# Lead Changes In Sports: An Analysis

No matter the sport, at any one time, either one of the competitors is winning or the scores are tied.

But have you ever wondered about how many times the lead changes per match?

## Probability

Many decisions are based on understanding how likely an event is to happen.

However, human instinct can get in the way of making the right decision.

Imagine a game where two equally skilled darts players are to play against one another.

How many times do you think the lead will change in the match?

Would you then expect this number to increase or decreased if they played more sets?

Well, since both players operate at the same skill level, the easiest way to determine to see how many times the lead will change is with a coin flip.

Start by allocating each player with heads or tails.

Now, for a change in the lead to take place, the player that falls behind at the first toss must overtake the other player.

Therefore, the next thing to do is to investigate how often equalization occurs ie when the scores become level again.

Let’s say you flipped the coin 6 times.

Instinctively, you would assume that flipping either 6 heads or 6 tails in a row is not very likely to happen.

Coin FlipsNumber Of Combinations
12
24
38
416
532
664

From 6 coin flips, the probability that it lands on either heads or tails every flip is 3.125‬%.

Also, we understand that while there is a 50/50 shot of achieving either heads or tails per flip, it doesn’t mean that the results will end with 3 heads and 3 tails.

Now then:

The actual probability of getting either 6 heads or 6 tails in a row is 31.25%.

This is 10x larger than the instinctive approach, representing a huge difference.

But this does not also mean that we can expect to flip an equal number of heads or tails if we did 6 tosses 3 times.

## Equalization

Ok, so how likely is getting an equal number of heads or tails?

First, note that at any stage in the process, either heads or tails leads or or there is an equal number of flips.

In order to achieve equality, there must be an even number of coin tosses.

As the number of tosses increases, common sense suggests that it becomes more likely to achieve equality.

This way of thinking is based on the law of averages.

Here’s the deal:

Statistics say otherwise.

In fact, this way of thinking couldn’t be more wrong.

There’s an interesting passage in John Haigh’s “Taking Chances” where he analyzes equalization of coin flips.

Here are the results.

Number Of Coin TossesChance Of EqualityProbability
21/250%
43/837.5%
65/1631.25%
835/12827.34%
1063/25624.6%

As you can see, this goes completely against the common sense approach: the probability of equalization decreases as the number of coin tosses increases.

If we were to toss the coin for a total of, say 20 times, how many tosses should we now expect equalization?

Well, it could be any of the 11 possible even-numbered solutions across the timeline, from the first 2 tosses up to the 20th.

But where along the timeline is the most likely to achieve equalization?

At the beginning?

Somewhere in the middle?

Towards the end?

Most people would assume the middle, but according to David Blackwell, as you approach the middle, the less likely the coin toss achieves equalization.

Here are his findings:

Time Since Heads Or Tails Were Last EqualProbability Of Equalization
0 or 2017.62%
2 or 189.27%
4 or 169.27%
4 or 167.36%
6 or 146.55%
6 or 146.55%
8 or 126.17%
106.06%

These results follow the same pattern as Haigh’s previous study.

So, if we don’t get equality from early tosses, it’s more likely we will have to wait for longer until we get the desired result.

## How Many Times Will The Lead Change?

Now we arrive at the following question:

How does all the above affect how often the lead changes?

Here is a table to show how likely a lead will change over 101 coin tosses.

Number Of Lead ChangesProbability Of Equalization
015.8%
115.2%
214%
312.5%
410.7%
58.8%
66.9%
75.2%
83.8%
92.7%
101.8%
112.6%

This shows that the lead doesn’t change more than 4 times over 60% of the time.

Further, the lead changes between 5 and 9 times only about a quarter of the time, while it will only change over 10 times around 5% of the time.

Interestingly, after half of the coin tosses, no equalization took place 50% of the time, meaning that if heads or tails had the lead at the halfway point remained ahead for the remaining coin tosses.

## Relating Coin Toss Lead Changes To Sports Betting

It should be a bit easier to see how this applies to sports investing.

Coin tosses show that when equally-skilled players or teams play against each other, there can be lengthy time spans where no equalization occurs and times where it can occur on several occasions in a shorter time span.

Equalization is more likely to occur at the start or the end of the match, rather than during the middle.

While we used darts as the example earlier, can this same logic also apply to soccer betting?

This will require some additional investigation before we can assume so.

The coin toss provides a more theoretical approach but it’s very much relevant for sports bettors.

Of course, with all this being said, the outcome of a sporting event is more complicated than a coin toss.