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Find the value of the constant \(k\) so that $$ f(x)= \begin{cases}k x^2\left(1-x^3\right), & 0

How many even two-digit numbers can be constructed out of the digits 3, 4, 5, 6, and 7? Assume first that you may use the same digit again. Next, answer this question assuming that you cannot use a digit more than once.

Four components are inspected and three events are defined as follows: A = 鈥渁ll four components are found defective.鈥� B = 鈥渆xactly two components are found to be in proper working order.鈥� C = 鈥渁t most three components are found to be defective.鈥� Interpret the following events: (a) B ? C. (b) B ? C. (c) A ? C. (d) A ? C.

Given that the probability of error in transmitting a bit over a communication channel is \(8 \times 10^{-4}\), compute the probability of error in transmitting a block of 1024 bits. Note that this model assumes that bit errors occur at random, but in practice errors tend to occur in bursts. Actual block error rate will be considerably lower than that estimated here.

Show that if event \(B\) is contained in event \(A\), then \(P(B) \leq P(A)\).

A lot of components contains \(0.6 \%\) defectives. Each component is subjected to a test that correctly identifies a defective, but about 2 in every 100 good components is also indicated defective. Given that a randomly chosen component is declared defective by the tester, compute the probability that it is actually defective.

A certain firm has plants A, B, and C producing respectively \(35 \%, 15 \%\), and \(50 \%\), of the total output. The probabilities of a nondefective product are, respectively, \(0.75,0.95\), and \(0.85\). A customer receives a defective product. What is the probability that it came from plant C?

In a party of five persons, compute the probability that at least two of the persons have the same birthday (month/day), assuming a 365-day year.

A communication channel receives independent pulses at the rate of 12 pulses per microsecond \(\left(12 \mu \mathrm{s}^{-1}\right)\). The probability of a transmission error is \(0.001\) for each pulse. Compute the probabilities of (a) No errors per microsecond (b) One error per microsecond (c) At least one error per microsecond (d) Exactly two errors per microsecond

Plot the reliabilities of a \(k\) out of \(n\) system as a function of the simplex reliability \(R(0 \leq R \leq 1)\) using \(n=3\) and \(k=1,2,3\) [parallel redundancy, TMR (triple modular redundancy), and a series system, respectively].