Question
A building's status panel receives three independently triggered alerts during its daily self-test. A fire alert occurs with probability \(\frac{1}{5}\), a water-leak alert with probability \(\frac{1}{4}\), and a low-battery alert with probability \(\frac{1}{10}\).
The panel shows only one message, using the priority order FIRE, then LEAK, then BATTERY. For example, if a leak and a battery warning occur together but there is no fire alert, the screen shows LEAK. If none occur, the display is blank.
Find the probability of each of the four possible screen messages. In particular, does a BATTERY message appear on 10% of tests?
Get hint
Hint 1
An alert is displayed only when it occurs and all higher-priority alerts fail to occur.
Check your answer
Solution
The FIRE message appears whenever the fire condition is active, regardless of the other two: \(P(\text{FIRE})=\frac{1}{5}=0.20\).
LEAK is displayed if there is a leak but no fire: \(P(\text{LEAK displayed})=\frac{4}{5}\cdot\frac{1}{4}=\frac{1}{5}=0.20\). BATTERY is shown only if the battery alert occurs with neither fire nor leak: \(P(\text{BATTERY displayed})=\frac{4}{5}\cdot\frac{3}{4}\cdot\frac{1}{10}=\frac{3}{50}=0.06\). Finally, the display is blank if all three underlying alerts are absent: \(P(\text{blank})=\frac{4}{5}\cdot\frac{3}{4}\cdot\frac{9}{10}=\frac{27}{50}=0.54\).
These outcomes are disjoint and exhaustive. Their total probability is \(\frac{10+10+3+27}{50}=1\), an enumeration check of the complete display mapping. The underlying battery event has probability \(\frac{1}{10}\), but it is hidden whenever a higher-priority alert occurs. Thus a visible 6% frequency does not imply the battery is faulty in only 6% of tests.
Assumptions
The three alert indicators are mutually independent Bernoulli variables.
Only the highest-priority active alert is displayed, and no alerts produces a blank display.
Final Answer
FIRE: \(\frac{1}{5}\); LEAK: \(\frac{1}{5}\); BATTERY: \(\frac{3}{50}\); blank: \(\frac{27}{50}\). No: BATTERY appears on 6% of tests.