Fundamental Bridge¶

Proposition¶

For any event $A$, let $\indicator_A$ equal one on $A$ and zero outside $A$. The fundamental bridge is

The expectation always exists because the indicator is bounded.

Interpretation¶

The fundamental bridge says that the probability of an event equals the expected value of its indicator. It lets every probability be written as an expectation, and every expected count as a sum of probabilities.

In the figure $\mathbb{P}(A)=0.3$$0.3$: the indicator equals $0$ with probability $0.7$, when $A$ did not occur, and $1$ with probability $0.3$, when it did. Treating the two bars as weights, their balance point, the expectation, lies at $0\cdot0.7+1\cdot0.3=0.3=\mathbb{P}(A)$$\mathbb{P}(A)$.

The proof is one line, since $\indicator_A$ is a Bernoulli variable with parameter $\mathbb{P}(A)$. The identity becomes useful together with linearity of expectation: if $X=\indicator_{A_1}+\cdots+\indicator_{A_n}$ counts how many of the events occur, then $\mathbb{E}[X]=\mathbb{P}(A_1)+\cdots+\mathbb{P}(A_n)$$\mathbb{P}(A_n)$, whether or not the events are independent. For $n$ tosses with heads probability $p$ this gives the binomial mean $np$. If $n$ letters are put into $n$ envelopes at random, letter $i$ lands in its own envelope with probability $1/n$, so the expected number of correctly placed letters is $n\cdot\tfrac{1}{n}=1$ for every $n$, although these events are dependent.

The bridge does not say that $\indicator_A$ equals $\mathbb{P}(A)$. The indicator only takes the values $0$ and $1$; $\mathbb{P}(A)$ is its average value, and it is usually not a value the indicator takes.

Key Results¶

• $\mathbb{E}[\indicator_A]=1\mathbb{P}(A)+0\mathbb{P}(A^c)=\mathbb{P}(A)$$\mathbb{P}(A)$.
• $\displaystyle \mathbb{E}[\sum_i\indicator_{A_i}]=\sum_i\mathbb{P}(A_i)$ for finitely many events, without independence.

Examples¶

• For an even result on a fair die, $\mathbb{E}[\indicator_A]=1/2$$1/2$.
• For $10$ components each failing with probability $0.02$, the expected failure count is $0.2$, even if failures are dependent.

Prerequisites¶

• Indicator Random Variable
• Expectation
• Probability Function
• Event

Related Concepts¶

• Linearity of Expectation
• Inclusion-Exclusion Principle
• Law of Total Probability
• Adam’s Law: Total Expectation

Sources¶

• Probability Theory for EOR · Week 4
• Nicky van Foreest · Probability Distributions · Refresher

Aliases: indicator bridge, expectation of an indicator Keywords: probability-expectation bridge Symbols: sum

Last updated: 20 October 2026 03:02