Among 6411 cases of heart pacemaker​ malfunctions, 340 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 6411 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? What is the probability? The procedure is what to result in the entire batch being accepted?

To find the probability that the firmware in the entire batch will be accepted, we need to first find the probability of no failures in the selected 3 pacemakers.

The probability of no failure in a single pacemaker is given by:

$$P(\text{No failure in a single pacemaker}) = 1 - P(\text{Failure in a single pacemaker})$$

Given that there are 6411 cases of heart pacemaker malfunctions and 340 were found to be caused by firmware, the probability of failure in a single pacemaker is:

$$P(\text{Failure in a single pacemaker}) = \frac{340}{6411}$$

Therefore, the probability of no failure in a single pacemaker is:

$$P(\text{No failure in a single pacemaker}) = 1 - \frac{340}{6411}$$

Now, we need to find the probability of no failure in 3 different pacemakers. Since these pacemakers are randomly selected, we use the multiplication rule for independent events. The probability of no failure in 3 different pacemakers is:

$$P(\text{No failure in 3 different pacemakers}) = \left(1 - \frac{340}{6411}\right)^3$$

So, the probability that the firmware in the entire batch will be accepted is:

$$\text{Probability}=\left(1-\frac{340}{6411}\right)^3=0.8492$$

This procedure is likely to result in the entire batch being accepted because probability 0.8492 is close to 1.