Question

Alexa has an important meeting in the morning. She sets three battery-powered alarm clocks just to be safe. If the first alarm clock has a $\frac{1}{{}^4{C}_1}$ probability of malfunctioning, second alarm clock has a $\frac{1}{{}^3{C}_1}$ probability of malfunctioning and the third alarm clock has a $\frac{1}{{}^5{C}_1}$ probability of malfunctioning, what is the probability that all the three alarm clocks fail at the same time?

• A. $\frac{1}{12}$
  
• B. $\frac{1}{15}$
  
• C. $\frac{1}{20}$
  
• D. $\frac{1}{60}$
  

Answer 1

The correct option is D $\frac{1}{60}$


The clocks are battery-powered we can assume that the failing of one clock will have no effect on the operation of the other two clocks.

The functioning of all the three clocks is independent.


Let $\text{P(F1)}$ denotes the probability of failing of first clock.
$\text{P(F1)}=\frac{1}{{}^4{C}_1}$
$\text{P(F1)}=\frac{1}{4}$


Let $\text{P(F2)}$ denotes the probability of failing of second clock.
$\text{P(F2)}=\frac{1}{{}^3{C}_1}$
$\text{P(F2)}=\frac{1}{3}$


Let $\text{P(F3)}$ denotes the probability of failing of third clock.
$\text{P(F3)}=\frac{1}{{}^5{C}_1}$
$\text{P(F3)}=\frac{1}{5}$

Let $\text{P(F1,F2,F3)}$ denotes the probability of failing of all the three clocks at the same time. This probability would be obtained by multiplying $\text{P(F1)}$, $\text{P(F2)}$ and $\text{P(F3)}$ as all these three events are independent events.

$\text{P(F1,F2,F3)}=\text{P(F1)}\times \text{P(F2)}\times \text{P(F3)}$

$=\frac{1}{4}\times \frac{1}{3}\times \frac{1}{5}$

$=\frac{1}{60}$

The probability of failing of all the three clocks at the same time is $\frac{1}{60}$.

Therefore, option (d.) is the correct answer.