Question

A scientist is doing study on person sleep. He classifies sleep activity in four stages: $1,2,3$ and $4$. Stage $3$ is the only stage considered to be deep sleep.

Ajay goes to clinic for sleeping brainwashes analyzed. Doctor monitored the person's brainwashes in $15-$minute intervals, for $8$ continuous hours, and categorized them into one of the four stages. The graph above shows this study.

If one $15-$minute time period is chosen at random, then the probability that the patient was in deep sleep during that time is $\frac{a}{b}$, where $a,b$ are co-primes. What is the value of $a+b$?

Answer 1

Total number of 15 minute interval in 8 hours $=\frac{8\times 60}{15}=32$.

Now from the graph we can observe that Ajay achieves Stage 3 sleep 12 times in 8 hours. Hence the required probability is $\frac{12}{32}=\frac{3}{8}$. Now 3 and 8 are co-primes. Hence $a=3$ and $b=8$. Therefore $a+b=11$.