Coin flips and climate

The weather is unusual, but is the climate truly different? How would we know?

I toss a fair coin 10 times. Which of these patterns is more likely than the other?

• HTHTHTHTHT
• HHHHHTTTTT
• TTTTTTHHHH
• HTTHHHTHTT

Now toss a fair coin nine times. I get HHHHHHHHH. What's the chance of getting T on the next toss?

The answer to the first question is that all of these outcomes are equally likely, though some seem odder to us than others. They all show five tails and five heads, the most common result of tossing a coin ten times. [1].

The answer to the second question is, of course 50%.

Now for the interesting question.

I toss a coin 100 times and I get 95 tails. What is the chance that the coin is fair [2]?

What if find one side of the coin is more magnetic than the other?

What if you inspect the rim and notice a color change from one side to the other?

Each of those three observations makes it less likely that the coin is fair. Taken together they strongly suggest the coin isn't fair.

We know that weather is not "fair". It is biased by climate.  If the distribution of weather events changes, we may infer that the climate bias is changing. If we have strong reason to suspect that atmospheric CO2 concentrations change climate, and we know CO2 is rising and weather events are changing, we have even more reason to suspect that climate is changing.

That's why we can say, beyond a reasonable doubt, that our climate is changing.

[1] Contemplation of these results doubtless leads to speculations on the arrow of time, Boltzman's brains, and the insanely unlikely probability of my certain existence. But that's not for today.
[2] Can I reject the null hypothesis of a fair coin, where a fair coin, tossed a very large number of times, will turn up heads and tails with equal frequency?