91Ó°ÊÓ

An electronic system is made up of two components connected in parallel. Consequently, the system fails only when both of the components fail. The probability the first component fails is \(0.05\) and, when this happens, the probability the second component fails is \(0.02\). What is the probability the electronic system fails?

A carton contains 20 computer chips, four of which are defective. Isaac tests these chips \(-\) one at a time and without replacement - until he either finds a defective chip or has tested three chips. If the random variable \(X\) counts the number of chips Isaac tests, find (a) the probability distribution for \(X\); (b) \(\operatorname{Pr}(X \leq 2)\); (c) \(\operatorname{Pr}(X=1 \mid X \leq 2)\); (d) \(E(X)\); and (e) \(\operatorname{Var}(X)\).

a) If the letters in the acronym WYSIWYG are arranged in a random manner, what is the probability the arrangement starts and ends with the same letter? b) What is the probability that a randomly generated arrangement of the letters in WYSIWYG has no pair of consecutive identical letters?

Let \(S\) be the sample space for an experiment \(\mathscr{E}\) and let \(A, B \subseteq \mathscr{P}\) with \(A \subseteq B\). Prove that \(\operatorname{Pr}(A) \leq \operatorname{Pr}(B)\).

If the letters in the word BOOLEAN are arranged at random, what is the probability that the two O's remain together in the arrangement?

Using the laws of set theory, simplify each of the following: a) \(A \cap(B-A)\) b) \((A \cap B) \cup(A \cap B \cap \bar{C} \cap D) \cup(\bar{A} \cap B)\) c) \((A-B) \cup(A \cap B)\) d) \(\bar{A} \cup \bar{B} \cup(A \cap B \cap \bar{C})\)

In alpha testing a new software package, a software engineer finds that the number of defects per 100 lines of code is a random variable \(X\) with probability distribution: $$ \begin{array}{c|cccc} \boldsymbol{x} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\ \hline \boldsymbol{P r}(\boldsymbol{X}=\boldsymbol{x}) & 0.4 & 0.3 & 0.2 & 0.1 \end{array} $$ Find (a) \(\operatorname{Pr}(X>1)\); (b) \(\operatorname{Pr}(X=3 \mid X \geq 2)\); (c) \(E(X)\); and (d) \(\operatorname{Var}(X)\).

Three missiles are fired at an enemy arsenal. The probabilities the individual missiles will hit the arsenal are \(0.75,0.85\), and \(0.9\). Find the probability that at least two of the missiles hit the arsenal.

a) Among the strictly increasing sequences of integers that start with 1 and end with 7 are: i) 1,7 ii) \(1,3,4,7\) iii) \(1,2,4,5,6,7\) How many such strictly increasing sequences of integers start with 1 and end with \(7 ?\) b) How many strictly increasing sequences of integers start with 3 and end with \(9 ?\) c) How many strictly increasing sequences of integers start with 1 and end with \(37 ?\) How many start with 62 and end with \(98 ?\) d) Generalize the results in parts (a) through (c).

For positive integers \(n, r\) show that $$ \begin{aligned} \left(\begin{array}{c} n+r+1 \\ r \end{array}\right)=&\left(\begin{array}{c} n+r \\ r \end{array}\right)+\left(\begin{array}{c} n+r-1 \\ r-1 \end{array}\right)+\cdots \\ &+\left(\begin{array}{c} n+2 \\ 2 \end{array}\right)+\left(\begin{array}{c} n+1 \\ 1 \end{array}\right)+\left(\begin{array}{c} n \\ 0 \end{array}\right) \\ =&\left(\begin{array}{c} n+r \\ n \end{array}\right)+\left(\begin{array}{c} n+r-1 \\ n \end{array}\right)+\cdots \\ &+\left(\begin{array}{c} n+2 \\ n \end{array}\right)+\left(\begin{array}{c} n+1 \\ n \end{array}\right)+\left(\begin{array}{c} n \\ n \end{array}\right) . \end{aligned} $$