Original practice · fully worked
Original variant: aquaculture alarm nights
An aquaculture sensor independently raises an alarm on each of seven nightly checks with probability 0.35. After the monitoring period, the log shows alarms on exactly two nights. Calculate the conditional probability that the first alarm was on the third night.
- A
0.0569 - B
0.1905 - C
0.2857 - D
0.2985 - E
0.3500
Answer and calculation
There are 21 equally likely pairs of alarm nights. Four pairs have night three first—night three paired with one of nights four through seven—so the probability is 4/21, approximately 0.1905, and choice B is correct.
Setup
Setup
Given exactly two alarm nights, the selected pair is uniform among all pairs of the seven night positions.
#{possible alarm-night pairs}=(27)=21 Model
Model
If the first alarm is on night three, nights one and two are not selected, night three is selected, and the other alarm can occur on any later night.
#{favorable pairs}=(14)=4 Compute
Compute
Form the conditional ratio from the favorable and total pair counts.
Pr(first alarm on night 3∣two alarm nights)=214 214=0.1904761905 Answer
Answer
The rounded conditional probability is 0.1905.
0.1905(B) 