Probability that seems to exceed 1 Suppose we have the following information based on independent events:

*

*The probability of contracting Alpha variant is .4

*The Delta variant is 50% more contagious than Alpha

*The Omicron BA1 variant is 50% more contagious than Delta

*The Omicron BA2 variant is 50% more contagious than Omicron BA1

Calculating this as-is would give the following probabilities:

*

*Probability of Alpha = .4

*Probability of Delta = .6

*Probability of Omicron BA1 = .9

*Probability of Omicron BA2 = 1.35

Since we know that the probability of an event cannot exceed 1, it is clear I am not thinking about this correctly.  Can you nudge me in the right direction?

Think about rolling multiple 6 sided dice. (And rolling a particular number on a dice would be a model for getting infected)
The probability to roll 3 or less with one single dice is 50%.
When you have $1.5 \cdot 1.5 \cdot 1.5 = 3.375 \approx 3 $  more dice then this probability to roll a 3 or less is now, 87.5% and not 150% percent.


The number of dice is a very simple toy-model for the probability to get sick. In reality you have many factors, the number of people you encounter, the probability that they are infectious, how long they are infectious, how much viral parts they carry around, how well the transmission is, etc.
In terms of the contagiousness it is often related to the $R_0$ value which is a combined value of all above and relates to the number of other people that get infected by another infected person. (And this value is estimated by observing and comparing the growth rate of different viruses and using measurements in test animals, laboratory and interviews that investigate contacts of positively tested people and how many of those contacts get infected)
Then the probability to get sick/infected might be much more subtle. For a highly contagious disease like COVID most people will eventually get infected one or more times. The contagiousness only influences how fast it happens and how steep the peak will rise. It is not so much about if you get infected, but more about when you get infected.
