Two Alarms on One Phone Behave Like One Alarm, and the Arithmetic Shows It

A second alarm lowers your chance of oversleeping only by the share of failure causes it does not have in common with the first. Reliability engineers call the shared share the common-cause fraction and write it β (beta), and for two alarms on the same phone it is nearly all of the total, so the second alarm buys very little.

Everything below is arithmetic on assumed rates, not measurement.

What does the IEC 61508 beta factor say about redundancy?

IEC 61508, the international standard for functional safety of electrical and programmable equipment, handles redundancy with a blunt instrument. Part 6, Annex D, splits each channel’s dangerous failure rate in two: a fraction β assumed to strike every redundant channel at once, and the remainder, 1 − β, that strikes each channel independently. The ranges commonly cited from the annex are 0.005 to 0.05 for programmable electronic equipment and 0.01 to 0.1 for field equipment. This post relies on those published summaries rather than the paywalled standard, and the standard applies the ranges to dangerous failure rates in industrial equipment, so using β for a per-night probability in a bedroom is an approximation.

For two channels that each fail with probability p on a given night, the chance both fail is:

P(both) = β·p + ((1 − β)·p)²

The first term is the shared failure. The second is the two independent failures landing on the same night, and it is tiny whenever p is small. So the first term decides almost everything.

Take p = 2%: one failed morning in 50, or 7.3 nights a year. The 2% is chosen so the arithmetic is readable. No measured failure rate for consumer alarms enters this post.

Why do two alarms on one phone share most of their failures?

Run down the ways a morning alarm fails and ask of each one whether a second alarm on the same phone survives it:

• A battery that died overnight, or a phone that was switched off. Both alarms are gone.
• A phone left in another room, on a charger that wasn’t live. Both gone.
• A wrong-day or wrong-AM/PM entry made at 11 p.m. Usually made once, then copied. Both gone.
• A shifted clock after travel or a clock change (the wall-clock log for 2026 and 2027 lists the minutes involved). Both alarms read the same clock.
• A background rule that stops the app from firing, a pattern Android’s battery optimisation produces. The alarm is an app; so is its twin.
• Volume or sound turned down. Both gone.
• The sleeper hearing it and not waking. Both gone, and so is every other sound-based alarm in the room.

Only an entry typo or a single toggled-off alarm hits one and spares the other. Nearly every cause on the list takes both down, so a rough sort puts about nine in ten on the shared side. That gives β = 0.9 for “two alarms, one phone”. The sort is a judgement call, and the sensitivity run below shows what a lower value does.

At β = 0.9: P = 0.9 × 0.02 + (0.1 × 0.02)² = 0.018004. That is 6.57 failed mornings a year, against 7.30 for one alarm. The second alarm removed 0.73 of them. The value sits far outside the standard’s own range, which tops out near 0.1, because the standard assumes the channels have been separated by design and these have not.

Is the sleeper a shared part too?

Every sound-based alarm, whichever device plays it, has to get through the same sleeper. Bruck, Ball, Thomas and Rouillard’s 2009 paper in the Journal of Sleep Research (18:196–203) tested nine signals on 39 young adults during stage 4 sleep and found that low-frequency square waves at 400 and 520 Hz woke people at significantly lower levels than higher-pitched alternatives; the fire-safety work on awakening signals is where this blog treats it in full. Pitch and pattern change how loud a sound must be to work, which improves p for every sound-based channel at once. They cannot diversify the channels. Two alarms with one ringtone are two attempts at one lock.

The sleeper therefore sets a floor even for channels with separate batteries and clocks. Suppose two in five of the 2% failures are “heard it, did not wake” and so hit any sound-based backup too. A battery clock in the hall then has β = 0.4, and the table below uses that.

What counts as an independent channel?

Four tests, in the order a channel fails them:

1. Separate power. A battery clock in another room does not drain when the phone does.
2. Separate settings and time source. A second person’s phone, or a clock nobody has configured, does not inherit your entry mistake.
3. Separate path to the sleeper. Sound, light, touch and a person at the bedside reach a sleeper through different routes. No study cited here ranks them against one another; they are separate by construction, not by measurement. A person who physically enters the room is the only channel in the table that does not depend on the sleeper hearing anything.
4. Separate consequence. Some channels wake no one. They record the failure: a friend who is told when you haven’t checked in changes what a missed morning costs, and the pressure applies before bed, when you choose where the phone sleeps. This arithmetic does not credit such a channel with preventing a single oversleep.

How much does each fix change the count?

Setup	β	Chance both fail on a night	Missed mornings a year	Roughly 1 in
One alarm	n/a	2.0000%	7.30	50
Second alarm, same phone	0.90	1.8004%	6.57	56
Phone plus battery clock in the hall	0.40	0.8144%	2.97	123
Phone plus a person who comes into the room	0.05	0.1361%	0.50	735
Fully independent channels (a limit, not a setup)	0.00	0.0400%	0.15	2,500

All rows use the 2% rate, and every β is a choice. Sensitivity for the same-phone row: at β = 0.8 the figure is 5.85 mornings a year, and at 0.6 it is 4.40. Even at 0.6, two alarms on one phone leave most of the original failures standing, because β·p is a floor that no copy of the same channel can dig under.

The battery clock moves the count less than the person at the bedside, and the reason is the sleeper: the clock fixes the device causes and leaves the human one. A person in the room does not need the sleeper to hear anything.

How can you estimate your own beta?

Count, don’t guess. Over the last year, list every morning the first alarm failed or nearly failed: a dead battery, a wrong-day entry, a silenced phone, an app that never fired, a sleeper who heard it and turned it off. Then ask of each one whether a second alarm on the same phone would have survived it. If nine of ten incidents would have taken both alarms down, the working β is 0.9. Repeat the exercise for the proposed backup: would the battery clock in the hall have survived the same ten incidents?

The list is more useful than the number. Most people find that the incidents cluster in two or three causes (power, settings, the sleeper), and each cluster points at a different fix: a clock on mains power for the first, a person or a different kind of device for the second, a consequence that is visible to someone else for the third.

Cost matters as well. A 2% chance of missing an optional workout and a 2% chance of missing a flight are the same probability and very different problems, so the backup worth building is the one for mornings where p × cost is largest.

Where does this analogy stop?

Nobody has estimated the bedroom’s β, so the 2% rate and the 0.9, 0.4 and 0.05 are all assumptions. Mornings are not equally costly either: the arithmetic treats a late start for a Saturday swim the same as a missed 6:10 flight. A person at the bedside is less independent than the table assumes if they sleep in the same building or follow the same schedule. And the formula treats failures as arriving once a night, though real mornings fail in clusters after a bad week.

The shape holds up regardless: the first term dominates, so the useful question about a backup is what it has in common with the first alarm, not how many there are.

FAQ

What beta factor should a home alarm setup use?

No standard assigns one to bedrooms, so any value is a judgement. Under the assumptions used here, moving from a second alarm on the same phone (0.9) to a phone plus a battery clock in the hall (0.4) cuts expected missed mornings from 6.57 to 2.97 a year, a drop of about 55%. Adding a person who physically comes into the room (0.05) takes it to 0.50. Each shared battery, charger, room, setting or time source argues for a higher value, and so does a sleeper who tends to sleep through sound.

How many alarms are enough?

The failure probability can never fall below beta times the single-alarm probability, however many copies you add. With beta at 0.4 and a 2% single-alarm rate, the floor is 0.8% a night, or 2.92 missed mornings a year. A third channel at that beta lands at about 0.8002% a night, still 2.92 mornings a year, so the third copy changes almost nothing.

Does the 0.9 for two alarms on one phone fit inside the standard’s range?

No. The ranges commonly cited from IEC 61508-6 Annex D top out around 0.1 for field equipment, because the standard assumes engineers have already separated the channels. Two alarms on one phone are barely separated at all, so a value near 0.9 is outside anything the standard would accept, and that gap is the point of the exercise.

Is a smartwatch a second device?

Only if it holds its own alarm. A watch that mirrors the phone’s alarm shares the settings and usually the time source, so it behaves like the same channel; a watch with its own alarm and battery shares neither, but it still shares the wearer and the sleep stage the wearer is in.