Among 6245 cases of heart pacemaker malfunctions, 272 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 6245 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?
The probability is $\square$. This procedure is likely to result in the entire batch being accepted.
(Round to three decimal places as needed.)
Answer
Final Answer: The probability that the firmware in the entire batch will be accepted is \(\boxed{0.875}\). This procedure is likely to result in the entire batch being accepted.
Step 1 ：First, we need to calculate the probability of a single pacemaker not malfunctioning due to firmware. This is equal to the number of non-faulty pacemakers divided by the total number of pacemakers. In this case, there are 5973 non-faulty pacemakers out of a total of 6245 pacemakers. So, the probability of a single pacemaker not malfunctioning is \(\frac{5973}{6245} = 0.956\).
Step 2 ：Next, since the pacemakers are tested independently, the probability that all three pacemakers tested do not malfunction is the cube of the probability that a single pacemaker does not malfunction. So, the probability that all three pacemakers do not malfunction is \((0.956)^3 = 0.875\).
Step 3 ：Final Answer: The probability that the firmware in the entire batch will be accepted is \(\boxed{0.875}\). This procedure is likely to result in the entire batch being accepted.
