Among 8846 cases of heart pacemaker​ malfunctions, 375 were found to be caused by​ firmware, which is software programmed into the device. If the firmware is tested in 3 different pacemakers randomly selected from this batch of 8846 and the entire batch is accepted if there are no​ failures, what is the probability that the firmware in the entire batch will be​ accepted? Is this procedure likely to result in the entire batch being​ accepted?

Accepted Solution

Step-by-step explanation:Probability, P = [tex]\frac{no. of favourable outcomes}{Total no. of possible outcomes}[/tex]Out of 8846 heart pacemaker malfunctions cases, caused by firmware cases are 375then, no. of cases not caused by firmware are:8846 - 375 = 8471Probability for three different pacemakers respectively is given by:P(1) = [tex]\frac{8471}{8846}[/tex] we select the next malfunctioned pacemaker from the remaining i.e., out of 8471, excluding the chosen malfunctioned pacemakerP(2|1) =  [tex]\frac{8470}{8845}[/tex] Therefore, the events are not independent of each other Now, if the selection is without replacement, thenP(3|1 & 2) =  [tex]\frac{8469}{8844}[/tex]Now, by general multiplication rule(as the events are not independent):P(none of the 3 are caused by malfunction) = [tex]\frac{8471}{8846}\times\frac{8470}{8845}\times\frac{8469}{8844}[/tex]= 0.9019P(none of the 3 are caused by malfunction)  = 90.19%The probability is high, therefore the whole batch will be accepted.