If we stick with your 1/44 assumption, we can then assume 50% chance that the following day will also be a record setting day (probably too low still but the math is easier). Your one week estimate would be (1/44)*(1/2)^6.

While @bloodfoot@programming.dev is right about me not being right, I’m not sure their implementation is correct either. I grabbed the data sheet from here, and have calculated the following probabilities. Now that I’m invested I really want to get this ‘right’.

This the work book I’m in.

I think its most fair to calculate this using the marginal probability of the next day being warmer for a given month. Some months are generally warming, some generally cooling, so it makes sense to base the marginal probability of the next day being warmer for a specific month (rather than all months). The probability of a given day in July being warmer than the previous day is 0.615. So pretty close to 50/50, but better to be precise. I think if we’re going to go this way, we can also give ourselves the liberty of not starting on July first, but rather then trend can start any time in July.

So now the math becomes: State one is defined as the probability of having a record warm day in July: 1/44 *days in July State two is defined as the probability the next day being warmer than the previous in July : 0.615

Which when I put it into this calculator, we get… .9981? This seems extremely low. So much so, I think either the markov chain is implemented wrong, or that the markov principal doesn’t apply here.

This would put the probability of a summer like the one as a one in 500-ish year occurrence. My understanding and read of the news is that this is an event that is on the scale of ‘has never happened before’.

Update:

So I am still not confident in the markov implementation but I put together another approach. I did a one sided T-test of this July versus all the previous Julys.

Here is a plot of that (month is on the X).

T-statistic came in too low to calculate a p-value.