Question

A fire-detection device uses three temperature-sensitive cells acting independently of one another so that any one or more can activate the alarm. Each cell has a probability \(p=.8\) of activating the alarm when the temperature reaches \(57^{\circ} \mathrm{C}\) or higher. Let \(x\) equal the number of cells activating the alarm when the temperature reaches \(57^{\circ} \mathrm{C}\). a. Find the probability distribution of \(x\). b. Find the probability that the alarm will function when the temperature reaches \(57^{\circ} \mathrm{C}\). c. Find the expected value and the variance for the random variable \(x\).

Short answer

Answer: The probability that the fire-detection device will function when the temperature reaches 57°C is 0.992. The expected value for the random variable representing the number of cells activating the alarm is 2.4, and the variance is 0.48.

Step-by-step solution

Calculate probability distribution of x

Since we have three cells and each cell can either activate or not activate the alarm, there are a total of 2^3 = 8 possible outcomes. The probability distribution can be calculated using the binomial distribution formula: \(P(x) = \binom{n}{x} p^x (1-p)^{n-x}\) Here, \(n = 3\) (total number of cells), \(x\) takes values from 0 to 3 (number of cells activating the alarm), and \(p = 0.8\). For \(x=0\): \(P(x=0) = \binom{3}{0} (0.8)^0 (1-0.8)^{3-0} = 1 (1) (0.2)^3 = 0.008\) For \(x=1\): \(P(x=1) = \binom{3}{1} (0.8)^1 (1-0.8)^{3-1} = 3 (0.8) (0.2)^2 = 0.096\) For \(x=2\): \(P(x=2) = \binom{3}{2} (0.8)^2 (1-0.8)^{3-2} = 3 (0.64) (0.2)^1 = 0.384\) For \(x=3\): \(P(x=3) = \binom{3}{3} (0.8)^3 (1-0.8)^{3-3} = 1 (0.512) (1) = 0.512\) The probability distribution of x is: \(P(x=0) = 0.008, P(x=1) = 0.096, P(x=2) = 0.384, P(x=3) = 0.512\)

Calculate the probability that the alarm will function

To find the probability that the alarm will function when the temperature reaches 57°C, we need to calculate the probability of \(at\ least\ one\) cell activating the alarm. We can also calculate the probability of no cells activating the alarm and subtract it from 1. \(P(at\ least\ one\ cell) = 1 - P(x=0) = 1 - 0.008 = 0.992\) The probability that the alarm will function when the temperature reaches 57°C is 0.992.

Calculate the expected value and variance of x

The expected value (mean) and variance for the random variable \(x\) in a binomial distribution can be calculated using the following formulas: \(E(x) = np\) \(Var(x) = np(1-p)\) Here, \(n = 3\) and \(p = 0.8\). \(E(x) = 3(0.8) = 2.4\) \(Var(x) = 3(0.8)(1-0.8) = 3(0.8)(0.2) = 0.48\) The expected value for the random variable \(x\) is 2.4, and the variance is 0.48.

Key concepts

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take on one of two independent outcomes under a given number of trials. In the case of the fire-detection device, each temperature-sensitive cell has two possible outcomes: it either activates the alarm or it does not.

When calculating the probability distribution for the number of cells that will activate the alarm, we use a formula specific to binomial distributions:
\[\begin{equation}P(x) = \binom{n}{x} p^x (1-p)^{n-x}\end{equation}\]
where:

• \(n\) is the number of trials, or cells, which is 3 in this example.
• \(x\) is the number of successes, or cells that activate the alarm.
• \(p\) is the probability of a single cell activating the alarm on a temperature of at least \(57^\circ \mathrm{C}\), which is 0.8.
• The term \(\binom{n}{x}\) is a binomial coefficient and represents the number of ways x successes can occur in n trials.

Each cell operates independently, which satisfies one of the key criteria for the trials in a binomial distribution.

Random Variable

A random variable is a numerical description of the outcome of a statistical experiment. In our fire-detection scenario, the random variable \(x\) represents the number of cells that activate the alarm when the temperature exceeds a certain threshold. Since \(x\) can only take on a countable number of finite outcomes (0, 1, 2, or 3), it is classified as a discrete random variable.

It is essential to distinguish between the random variable \(x\) and the possible outcomes it can have. The variable itself is not the outcome, it's the feature we are observing, which in this case is the activation count of the temperature-sensitive cells.

Expected Value

The expected value, often denoted as \(E(x)\), is more commonly known as the mean and provides a measure of the center of the distribution of the random variable. This value gives us the average result if the random process could be repeated an infinite number of times. In the binomial distribution framework, the formula for the expected value is given by:
\[\begin{equation}E(x) = np\end{equation}\]
For the fire-detection device example, the calculation is as follows:
\[\begin{equation}E(x) = 3 \times 0.8 = 2.4\end{equation}\]
This result means that, on average, we can expect 2.4 cells to activate the alarm when the temperature reaches \(57^\circ \mathrm{C}\) or higher. It is a measure of the system's reliability in the given conditions.

Variance

Variance is a measure of the spread of a probability distribution. It quantifies how much the values of the random variable \(x\) deviate from the expected value \(E(x)\). For a binomial distribution, the variance is calculated using the formula:
\[\begin{equation}Var(x) = np(1-p)\end{equation}\]
which for our fire-detection system becomes:
\[\begin{equation}Var(x) = 3 \times 0.8 \times (1 - 0.8) = 0.48\end{equation}\]
The variance of 0.48 indicates that there is some variability in the number of cells that will activate the alarm at the specified temperatur blockIdx. A higher variance would suggest a less reliable system, whereas a variance of zero would imply no variation, indicating perfect reliability. Understanding the variance helps in assessing the consistency of the alarm's performance.