Probability of drawing a sequence of increasing large outliers

Question

News today (e.g, NY Times article) that the Earth has set a third successive high temperature record made me wonder if the temperatures were not increasing, what would be the probability of such an outcome.

Here's an attempt at formalizing that question: suppose that global mean temperature is normally distributed $N(\mu, \sigma)$ and $N$ samples have been drawn from this distribution: $T_1, \cdots, T_N$. Let $T_{maxN}$ be the maximum of those first $N$ samples.

What then is the probability that the next three samples drawn are successively greater than the maximum of the first $N$ samples, i.e.,

$$P(T_{N+3} > T_{N+2} > T_{N+1} > T_{maxN})$$

Answer 1

If there have been $n$ items, with the value of each randomly chosen from the same distribution, the chance of the $n+1$-st being largest is $\frac1{n+1}$.

Repeating this, the chance of the three consecutive records is $\frac1{(n+1)(n+2)(n+3)}$.

Answer 2

That's just like asking:

"if there is a race between $N+3$ identical contestants $1,2,\ldots N+3,$ with equal chance of finishing in any order, what is the probability that the first three finishers are contestants $1,2,3$ in that order."

There are $N!$ different orders satisfying this criterion (from the $N!$ rearrangements of the finishers from 4th place to last) and $(N+3)!$ orderings total, so the probability is $$\frac{N!}{(N+3)!} = \frac{1}{(N+1)(N+2)(N+3)}$$

Answer 3

Reality check. Here is a simulation of your version of the question with only $n = 5$ historical observations. (With larger $n,$ chances pretty clearly go down.) Without loss of generality, assuming there is no temperature trend, I took all observations to be standard normal.

My variable $X$ counts the number of the three new observations that are above the maximum of the five old ones. I got $P(X = 3) \approx 0.018.$ So just that is a rare event (regardless of the order of the three new temperatures.)

My random variable $Y$ counts the number of increases in the sequence $(T_{5:5}, T_6, T_7, T_8),$ where $T_{5:5}$ is the biggest of the historical five observations. In order for $T_{5:5} < T_6 < T_7 < T_8$ we must have $Y = 3,$ and that never happened in a million iterations.

m= 10^6; n = 5; x = y = numeric(m)
for (i in 1:m) {
  t = rnorm(n); t.max = max(t)
  new = rnorm(3)
  x[i] = sum(new > t.max)                # nr new above t.max
  y[i] = sum(diff(c(t.max, new) > 0))  } # 3 means incr from t.max on
mean(x == 3);  mean(y == 3)
## 0.017736  # aprx prob all 3 new exceed max of 5 historical
## 0         # aprx prob max.t < t.6 < t.7 < t.8

Certainly, this doesn't mean temperatures aren't trending upward, by some reasonable definition of 'trending upward'. It just means the particular event NYT chose to report is rare, if temperatures are steady. (Newspapers are partly in the business of reporting surprising events.) Maybe you need to formulate and investigate some scheme such as the one @N74 suggests.

Note: Driving around with a friend earlier today, we noted digits of license plates of five oncoming cars (part of an extraordinarily stupid game). In our state most plates have a cluster of three digits: We recorded 873 129 322 573 903, what are the chances that will be exactly repeated on our next trip?