Among 7721 cases of heart pacemaker malfunctions, 336 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 7721 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 (Round to three decimal places as needed.)
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Let's break this down and find the probability that all three selected pacemakers have firmware that is functioning properly. First, we calculate the probability of selecting a functioning firmware pacemaker. Out of 7721 cases, 336 are faulty due to firmware. Hence, the number of functioning pacemakers is: \[ 7721 - 336 = 7385 \] Now, the probability that one randomly selected pacemaker has functioning firmware is: \[ P(\text{functioning}) = \frac{\text{number of functioning pacemakers}}{\text{total pacemakers}} = \frac{7385}{7721} \] Now, since we are selecting 3 pacemakers and we want all of them to function properly, we can raise the individual probability to the power of 3 (assuming independence): \[ P(\text{all 3 functioning}) = \left( \frac{7385}{7721} \right)^3 \] Calculating that gives: \[ \frac{7385}{7721} \approx 0.95746 \] Now raising to the power of 3: \[ P(\text{all 3 functioning}) \approx (0.95746)^3 \approx 0.876 \] So the probability that the firmware in the entire batch will be accepted is approximately: \[ 0.876 \] This means there's about an 87.6% chance that the batch will pass the test when selecting 3 random pacemakers. As for whether this procedure is likely to result in the entire batch being accepted, with such a high probability of passing the test, it appears reasonably likely to accept the entire batch without failure.
