Question

Among 81678167 cases of heart pacemaker​ malfunctions, 457457 were found to be caused by​ firmware, which is software programmed into the device. if the firmware is tested in 33 different pacemakers randomly selected from this batch of 81678167 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

Answer 1

(Note: This seems when question was pasted from its source the in text box, the numbers were pasted twice. Solving the question considering only one occurence of number i.e using 8167 instead of 81678167. However, method will be same regardless of the numbers listed)

Part a)
Total number of cases = 8167
Number of Malfunctions due to firmware = 457

Probability that a malfunction is because of firmware = $\frac{457}{8167}$

Probability that the malfunction is not because of firmware = 1 - Probability that a malfunction is because of firmware = $1- \frac{457}{8167}= \frac{7710}{8167}$

Probability of selecting 3 pacemakers without a malfunction because of firmware = $( \frac{7710}{8167} )^{3} = 0.841$


Part b)
The probability of selecting 3 pacemakers without any malfunction because of firmware is 0.841, which is a high probability. So we can say, yes this procedure is very likely to result in entire batch being accepted. The probability near 1 is considered a very higher probability for an event and the probability near 0 is considered a very lower probability for an event.