Question

Firefighters’ use of gas detection devices. Two deadly gasesthat can be present in fire smoke are hydrogen cyanide and carbon monoxide. Fire Engineering (March 2013) reported the results of a survey of 244 firefighters conducted by the Fire Smoke Coalition. The purpose of the survey was to assess the base level of knowledge of firefighters regarding the use of gas detection devices at the scene of a fire. The survey revealed the following: Eighty percent of firefighters had no standard operating procedures (SOP) for detecting/monitoring hydrogen cyanide in fire smoke; 49% had no SOP for detecting/monitoring carbon monoxide in fire smoke. Assume that 94% of firefighters had no SOP for detecting either hydrogen cyanide or carbon monoxide in fire smoke. What is the probability that a firefighter has no SOP for detecting hydrogen cyanide and no SOP for detecting carbon monoxide in fire smoke?

Short answer

The probability is 0.35.

Step-by-step solution

Given information

According to the information,

A=No. SOP for detecting hydrogen cyanide in fire smoke.

B=No. SOP for detecting carbon monoxide in fire smoke.

$$\begin{gathered} P\left(A\right)=80\%=0.8 \\ P\left(B\right)=49\%=0.49 \\ P\left(A\mathrm{or}B\right)=94\%=0.94 \end{gathered}$$

The formula for probability$\mathbf{P}\frac{\mathbf{Number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{favourable}\mathbf{}\mathbf{out}\mathbf{}\mathbf{comes}}{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{out}\mathbf{}\mathbf{comes}}$.

The general addition rule for two events$\mathbf{P}\left(\mathrm{AorB}\right)\mathbf{=}\mathbf{P}\left(A\right)\mathbf{+}\mathbf{P}\left(B\right)\mathbf{-}\mathbf{P}\left(A\mathrm{andB}\right)$.

Find the probability

$$\begin{gathered} P\left(A\right)=80\%=0.8 \\ P\left(B\right)=49\%=0.49 \\ P\left(AorB\right)=P\left(A\right)+P\left(B\right)-P\left(AandB\right) \\ P\left(AandB\right)=P\left(A\right)+P\left(B\right)-P\left(AandB\right) \\ =0.8+0.49-0.94 \\ =0.35 \\ =35\% \end{gathered}$$

Therefore,the probability is 0.35