No. The odds of the backup transmitter failure are exactly the same: they are not dependent events. If your probability of backup failure is "astronomical", then so is the probability of primary failure. .
I suppose you carry a lot of spare tires in your car? Of course not, no one does. Not because we are all defying logic and common sense. It's because the odds are certainly NOT the same as here we are talking about the probability of failure
over time.
Over 100s of dives, 100s of hours of dive time, the odds of a computer, or any piece of gear- failing, are astronomically higher than the probability of failure during what might be at most a 30 minute window- the remainder of a dive following failure of a primary piece of dive gear, in this case a dive computer.
Here's another way of looking at the probabilities, for those of us who are statistically challenged.
There's a woman who works for me, and when she goes on vacation she expresses her fear of flying and says "I know it sounds morbid but I almost wish there was a plane crash the day before my trip because what are the odds of TWO plane crashes in the same week?"
Now of course her logic is flawed, the odds of a second plane crash do not change just because there's a first (unless of course there are safety bulletins issued, aircraft taken out of service, maintenance issues that are addressed, etc, that do in fact decrease the odds of another failure but lets not go there as it is not relevant to this particular scope of discussion).
The odds of 2 transmitters failing are astronomically smaller than one, (just like the odds of two major plane crashes occurring at the same time are astronomically small), even though a failure of the first does not change the odds of the second, just like throwing a coin that lands heads doesn't make it any more likely that the next toss will be tails, even though OVER TIME the odds approach 50/50.
I hope this clears up the misconception on this thread regarding the odds of 2 failures being the same as one.