Run Expectancy and the 24 Base-Out States

By 4Baseball Staff · November 5, 2026 · 13 min read

Run expectancy gives every moment of an inning a value in runs. This explainer builds the 24 base-out states, shows how a table is made, and uses worked examples to test the sacrifice bunt and the stolen base.

Every half-inning of baseball can be described by two facts: who is on base and how many outs there are. Three bases can each be occupied or empty, which gives eight possibilities, and there can be zero, one or two outs, which gives three. Multiply them and you get 24 base-out states, and each of them has a value that can be measured: the average number of runs a team goes on to score from that point until the third out. That value is run expectancy, and it is one of the most useful ideas in the modern study of the game.

The idea sounds technical, but it is a way of putting numbers on instincts every fan has. Everybody knows that a runner on third with nobody out is a better situation than a runner on first with two outs. Run expectancy says how much better, in runs, and it lets you compare things that look unrelated: a walk, a stolen base, a sacrifice bunt, a double play. All of them are moves from one state to another, and the difference in value between the two states is what the move was worth.

This article builds the table from scratch, shows how a team's or a player's contribution is measured with it, and then applies it to the two most argued-over small-ball plays, the sacrifice bunt and the steal. It uses no table of measured values, because the numbers change with the era, the league and the ballparks; it uses the structure, the formulas and worked examples built on clearly labeled illustrative figures.

The 24 states

Start with the bases. Each base is either occupied or empty, so there are two choices for first, two for second and two for third, which makes eight base states. Listing them from the emptiest to the fullest gives this set.

Now add the outs. Each of the eight base states can occur with no outs, one out or two outs, so there are 8 times 3, or 24 states. Three outs ends the half-inning and is not a state in the table; its value is zero, because no more runs can be scored. When people say "the 24 base-out states," they mean this grid.

The ordering of the table follows from logic and not from measurement. More runners on base means a higher value, because every extra runner is an extra chance to score. More outs means a lower value, because each out removes a chance to keep the inning alive. So the lowest value in the grid is bases empty with two outs, and the highest is bases loaded with no outs. The numbers between those corners are what a table of run expectancy measures. In recent decades, no outs and nobody on has been worth roughly half a run, and bases loaded with no outs has been worth more than two, but the exact values depend on the league and the year.

How a table is built

The method is simple enough to do with a spreadsheet and a season of play-by-play data. For every plate appearance in a season, record the base-out state at the start of the plate appearance. Then look forward to the end of that half-inning and count the runs the team scored from that moment, including any scored on the plate appearance itself. Group the plate appearances by state and take the average of the runs scored in each group. That average is the run expectancy of the state.

Here is a hypothetical to make it concrete. Suppose that over a season a league had 200 plate appearances that began with a runner on first and nobody out. In the rest of those half-innings, the batting teams scored 170 runs in total. The run expectancy of that state is 170 divided by 200, or 0.85 runs. Do the same for the other 23 states and you have a table.

The important detail is that the count includes everything that happened afterward, so the value of a state already contains all the possible futures: the single that scores a runner, the double play that ends the threat, the strikeout, the home run. Run expectancy averages over the whole menu. It does not know whether a particular hitter at the plate is strong or weak, and it does not know whether a particular defense is good, so it is a statement about the average situation. That is both its strength, because it is simple, and its main limit, because it ignores who is playing.

This idea has a history. George Lindsey published an analysis of baseball strategy in the journal Operations Research in 1963, working out the value of different situations from a sample of games. Pete Palmer and John Thorn popularized the approach in their 1984 book The Hidden Game of Baseball, which turned it into a measure of how much a batter's actions were worth in runs. The 2007 book The Book by Tom Tango, Mitchel Lichtman and Andrew Dolphin brought the tables to a wide audience and used them to examine strategy. Since then the idea has become part of the standard vocabulary of the sport.

Measuring a play: from run expectancy to RE24

A table of states leads directly to a way of scoring individual plays. Every event moves the game from one state to another and may score some runs along the way. The value of the event is the change in run expectancy plus the runs that scored on it:

value of a play = RE after the play − RE before the play + runs scored on the play

When that value is added up for everything a batter does in a season, the total is called RE24, because it uses the 24-state table. The sign is the part to remember. A good play makes the value positive and a bad one makes it negative, and the size of the number tells you how good or bad.

The easiest case needs no table at all. A solo home run with the bases empty leaves the bases empty with the same number of outs, so RE after equals RE before, and the value of the play is just the one run that scored. The formula gives exactly +1.0. That makes sense: a solo home run is worth one run whether it comes in the first inning or the ninth. A three-run home run with two runners on also leaves the bases empty, so the value is 3 plus the change in expectancy, and that change is negative because the two runners were carrying some of the value before the swing. The batter is credited with 3 runs minus the potential those runners already held, which is a reminder that some of the credit belonged to the men who got on base ahead of him.

The ordinary out shows the other side. A strikeout with nobody on and nobody out moves the game from the empty, no-out state to the empty, one-out state. Nothing scored, so the value is the drop in expectancy between those two states, a small negative number. It is small because the situation was not very valuable to begin with. The same strikeout with the bases loaded and nobody out moves the game from the highest-value state to a lower-value state, and the drop is much larger. This is why a strikeout with runners on is more costly than one with the bases empty, a fact the batting average ignores.

Linear weights and where the idea leads

Averaging the change in run expectancy caused by each type of event, across a whole season, gives the run value of that event. A single is worth a certain amount on average, a double is worth more, a walk is worth less than a single and a home run is worth the most, and an out is worth a negative amount. These average values are called linear weights, and they are the basis of several modern measures of batting. A hitter's total contribution can be estimated by multiplying the number of singles, walks and so on by their weights, and the sum tells you how many runs above an average hitter he produced.

This has two practical consequences. The first is that the relative values of events are not the same as the way traditional statistics count them. Batting average treats a single and a home run alike as a hit, and treats a walk as nothing, but run values show that a walk is worth a good deal more than zero and a home run is worth far more than a single. The second is that the weights themselves depend on the run environment. In a season where runs are plentiful, each baserunner is more likely to come around to score, so walks and hits are worth somewhat more and outs cost somewhat more than in a low-scoring one. Run values of this kind are also the batting building block of the measures described in what WAR counts and cannot count.

Testing the sacrifice bunt

A sacrifice bunt trades an out for a base. With a runner on first and no outs, a successful bunt typically leaves a runner on second with one out. The question is whether the new state is worth more than the old one, and the table answers it by comparison.

Take illustrative numbers, invented for the example and not measured: suppose the runner-on-first, no-out state is worth 0.90 runs and the runner-on-second, one-out state is worth 0.70. The bunt, if it works perfectly, lowers the expectancy by 0.20 runs. The reason is simple. The out used up a third of the half-inning's supply of outs, and the extra base did not make up for it. In most run environments that have been studied, the published tables show this pattern: the bunt gives up more than it gains, in the average situation, when the goal is to score as many runs as possible over the whole inning.

That is not the end of the argument, and it is important to see why. Run expectancy measures the average number of runs. Sometimes a team does not want the average; it wants a single run, because the game is tied in the late innings and one run wins it. The probability of scoring at least one run is a different quantity from the average number of runs, and in a low-scoring environment a runner on second with one out can have a better chance of scoring once than a runner on first with none, even though the average number of runs is lower. The right tool for that decision is win probability and leverage, which looks at the score and the inning as well as the state.

Other factors also matter. The table assumes an average batter. If the hitter at the plate is a pitcher who is very unlikely to reach base, the bunt costs less because the alternative is weaker. If the defense is poor at fielding bunts or the batter is fast, the odds of a hit rise. And a bunt that fails, a pop-up or a force at second, costs more than the successful sacrifice gains. The long-standing finding from the tables is not that the bunt is always wrong but that the right situations are narrower than habit suggests.

Testing the stolen base

A steal has two outcomes, and the table gives a value to each. If the runner is safe at second, the game moves from the runner-on-first state to the runner-on-second state with the same number of outs: a gain. If he is caught, the game moves from the runner-on-first state to the bases-empty state with one more out: a loss, and usually a bigger one.

Call the gain G and the cost C. If the runner succeeds with probability p, the expected change is p times G minus (1 − p) times C. That is zero when p equals C divided by G plus C, and that value of p is the break-even success rate. Above it, the steal helps on average; below it, it hurts.

With illustrative numbers again, suppose a safe steal adds 0.20 runs and a caught steal costs 0.45. The break-even rate is 0.45 divided by 0.65, or about 69 percent. A runner who succeeds 80 percent of the time would produce an average gain, and a runner who succeeds 60 percent of the time would produce an average loss. Published estimates of the break-even rate have commonly landed somewhere in the neighborhood of 70 to 75 percent, which is why a runner with a good success rate is valuable and a runner who is caught often is a liability even if he steals many bases. The rate is not fixed; it changes with the number of outs, the run environment and who is batting.

This is also why raw stolen-base totals can mislead. A player with 40 steals and 20 times caught has a success rate of two-thirds, which is probably below break-even. A player with 25 steals and 3 times caught has about 89 percent and almost certainly helped his team more. The first has the bigger number on the leaderboard, and the second has the better effect on the scoreboard.

What the table does not know

Run expectancy is a tool with limits, and a careful reader should know them.

• It averages across everyone. The value of a state is the same whether the batter is a star or a weak hitter and whether the pitcher is an ace or a reliever. Adjusting for the players involved is possible but requires more data.
• It ignores the score and the inning. A runner on third in a blowout and in a one-run game in the ninth are the same state to the table, but not to the manager. That is the gap that win probability fills.
• It depends on the era and the park. A table built from one season and one league applies cleanly to that league, and ballparks that favor hitters or pitchers shift the numbers, as the explanation of park factors shows. Using it in a different run environment, such as a college league with aluminum bats or a youth league, changes the numbers, though the shape of the table stays the same.
• It credits the batter for what the runners do next. In RE24, a single that sends a runner from first to third gets more credit than one that moves him one base, even though the runner's speed or an outfielder's arm may have decided it. The measure describes what happened; it does not assign cause.
• It cannot see defense directly. A great catch that turns a double into an out shows up as a change in state but not as a credit to the fielder, unless extra work is done.

None of that makes the tool less useful. It makes it a starting point for a conversation, which is what any good statistic should be.

Using the idea in your own scorebook

You do not need a database to apply the idea. Keep score for a few games, using the scoresheet page if you want a template, and write down the base-out state at the start of each plate appearance. After a few dozen games you can compute your own rough run expectancy for a youth or recreational league, which will be much higher than the major-league values in a league where walks and errors are common. The stat calculator helps with the arithmetic of rates.

The goal is not precision. It is to build a habit of asking what a decision costs and what it gains, in runs, before making it. It is the same habit that run expectancy teaches at the highest level.

Common questions

How many base-out states are there? There are 24: eight base configurations times three out counts. Three outs ends the half-inning and has a value of zero.

What is the highest-value state? Bases loaded with no outs, and the lowest is bases empty with two outs.

Is the sacrifice bunt always a mistake? No. It tends to lower the average runs expected, but it can raise the chance of scoring a single run, which matters in some late-game situations, and it can be right with a weak hitter at the plate.

What is the break-even rate for a stolen base? It equals the cost of being caught divided by the gain from success plus the cost. Published estimates have commonly been around 70 to 75 percent, and it varies with the situation.

Where can I find a current table? Baseball statistics sites that publish run expectancy by season usually give the 24 values for each year. Use a table from the same era and league as the question you are asking.

Where to go next

For the related ideas, read about Pythagorean expectation, which links runs to wins at the team level, and win probability and leverage, which adds the score and the inning. Aging curves show how the skills behind these numbers change over a career. To see the leaders in the traditional categories, use the leaders pages.