Problem 2 - An Introduction to Mathematical Statistics and Its Applications

Question

In a nuclear reactor, the fission process is controlled by inserting special rods into the radioactive core to absorb neutrons and slow down the nuclear chain reaction. When functioning properly, these rods serve as a firstline defense against a core meltdown. Suppose a reactor has ten control rods, each operating independently and each having an \(0.80\) probability of being properly inserted in the event of an "incident." Furthermore, suppose that a meltdown will be prevented if at least half the rods perform satisfactorily. What is the probability that, upon demand, the system will fail?

Step-by-Step Solution

Answer

The probability that, upon demand, the system will fail can be calculated by subtracting the probability of system success (the probability of at least half the rods working properly) from 1. This involves calculating the binomial probability for k successes given n trials and a success probability p, and then summing these probabilities for k = 5, 6, 7, 8, 9, 10.

Step 1: Understand the Binomial Distribution

The binomial distribution is defined by two parameters: the number of trials n, and the probability of success p on any given trial. The binomial probability is given by \(P(X=k) = C(n, k) * p^k * (1-p)^{n-k}\), where X is the number of successes, k is the number of successes we are interested in, C(n, k) is the number of combinations of n items taken k at a time, and p is the probability of success on any given trial.

Step 2: Calculate the Probability of System Success

To calculate the probability of system success, we need to find the probability of at least half the rods (5, 6, 7, 8, 9, or 10) working properly. This is equivalent to summing up the binomial probabilities for k = 5, 6, 7, 8, 9, 10.

Step 3: Calculate the Probability of System Failure

The probability of system failure is 1 minus the probability of system success. That is, if we let S denote the event that the system succeeds and F denote the event that the system fails, then \(P(F) = 1 - P(S)\). In other words, to find the probability of system failure, we subtract the probability of system success from 1.

Key Concepts

Probability of System Success

To determine the probability of system success, we need to understand the key concept of the binomial distribution. In our scenario, we are dealing with a nuclear reactor which has 10 control rods. Each rod has an individual probability of 0.80 for being properly inserted during an incident. For the reactor system to succeed and avoid a meltdown, at least 5 of these rods need to function correctly.

The situation is aptly modeled by a binomial distribution, which offers a method to calculate probabilities based on a series of independent trials. Here, each control rod is a single trial with two potential outcomes: success or failure. The probability of success for each rod is 0.80, and we aim to find the combined probability for cases where 5, 6, 7, 8, 9, or all 10 rods are successfully inserted.

In mathematical terms, we express this as the sum of probabilities from 5 to 10 using the binomial probability formula, which allows for calculations of such probabilities across discrete values. To find this sum, we evaluate:

• At least 5 rods: Find stepwise for each number of rods from 5 to 10.
• Use the formula: \[P(X = k) = \binom{n}{k} (p)^k (1-p)^{n-k}\]

By successfully applying this formula to each relevant outcome, we can add these probabilities up to find the overall probability of system success. This step is crucial before moving on to consider possible system failure.

Probability of System Failure

Once the probability of system success has been established, the probability of system failure can be easily calculated. It simply complements the system success probability because there are only two possible outcomes for the system - it either succeeds or fails.

From basic probability principles, we know that the sum of probabilities of all possible outcomes in a given scenario must equal 1. Therefore, to find the probability of system failure, we subtract the probability of system success from 1, following the formula:

\[P(F) = 1 - P(S)\]
Where:

• \(P(F)\) represents the probability of failure,
• \(P(S)\) represents the probability of system success.

This operation gives a clear view of how likely it is for the system to not perform as required under given circumstances. Understanding this concept is essential for assessing the risks and reliability in nuclear reactor systems, where safety is of paramount importance.

Nuclear Reactor Control Rods

Control rods are critical components within nuclear reactors. They play a crucial role in ensuring the reactor's safe operation by controlling the rate of the nuclear fission reaction.

In the context of nuclear reactors, control rods are designed to absorb neutrons. Neutrons are particles that propagate the nuclear chain reaction. By absorbing them, the control rods slow down this reaction, thereby preventing potential overheating or a reactor "incident."

Each control rod functions independently, meaning the failure or success of one does not directly impact others. This independence makes the binomial distribution a suitable model for calculating probabilities related to control rods during reactor incidents.

Control rods are essential for managing the balance between reactivity and safely shutting down the reactor in an emergency. Their proper insertion during an incident is critical to prevent a core meltdown. Therefore, understanding the success and failure probabilities of these rods is vital for assessing reactor safety and designing systems that can reduce the likelihood of catastrophic failures.