91Ó°ÊÓ

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: \(P(x=0) = 0.008, P(x=1) = 0.096, P(x=2) = 0.384, P(x=3) = 0.512\)

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.

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.