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This is the very first installment of a new feature in Plus: a column by David Spiegelhalter exploring issues of risk and uncertainty that concern all of us but confound m

The patch from figure 14 contains decagons and pentacles divided up as follows:

Each of these two substitutions introduces rhombi. These can be swallowed up by the two new shapes introduced here:

First let's calculate the frequency of a particular digit $i$ with $ i \neq 0$.

Its frequency among 1-digit numbers is clearly 1/9.

First note that there are 10¹⁰ possible combinations of the numbers from 1 to 10: there are 10 choices for the number that goes in the first position, 10 choices for the number that goes in the second position and so on. Since each digit is equally likely, each of these combinations is equally likely, so the chance of picking an individual combination is 1/10¹⁰.

Imagine picking the four hymn numbers out of a hat. First note that four-hymn combinations with one 1-digit number and three 3-digit numbers come in four types: the one digit number can occur in first, second, third or last place of the selection. So the overall chance of picking such a combination is equal to: