A homeowner has two smoke detector alarms installed, one in the dining room (adjacent to the kitchen) and one in an upstairs bedroom (above the kitchen). If cooking produces smoke in the kitchen, the probability of setting off the dining room alarm (D) is .95. The probability of setting off the bedroom alarm (B) is .40. The two alarms detect smoke independently of each other. If there is smoke in the kitchen, what is the probability that the smoke will be detected and will set off an alarm?
Let's first understand what probability we need to find: The smoke from the fire is detected if it sets off the dining room alarm (D) or the bedroom alarm (B) or both, and therefore P(smoke is detected) = P(D or B), which is the probability we need to find. To that end, we are given a few pieces of information. Let's summarize them:
To that end, we are given a few pieces of information. Let's summarize them:
- P(D) = .95
- P(B) = .4
- Unlike the previous examples, in which P(A and B) was simply given, here we have a different piece of information: "The two alarms detect smoke independently of each other." In other words, instead of being given P(D and B), we are given the fact that D and B are independent.
Hint 1
Explain why we have enough information to find P(D and B); then find P(D and B).
Now use the information you are given and the General Addition Rule to find the desired probability, P(smoke is detected).