Question

2. Among 150 people interviewed as part of an urban mass transportation study, 84 ive more than Skm from the c ity centre (4), 99 regularly drive to work (B), 85 would gladly switch to public mass transportation if it were available (C), 74 are both A and B. 8 are A and C only, 9 are B and Conly, and 27 are none of the three categories. Find 3. At an electronic plant, it is known from past experience that the probability that a new employee who has attended the company's training program will meet the production quota is 0.86, and that the corresponding probability for a new employee who has not attended the company's training program is 0.30. If 80% of all new employees attend the training program, what is the probability that a new employee will meet the production quota? 4. A quality engineer inspects a random sample of 3 batteries from each lot of 24 car batteries that is ready to be shipped A. If such a lot contains 6 batteries with slight defects, what is the probability that a lot will contain at least 2 defective batteries? B. If a lot contains any defective batteries it will not be shipped. If the quality engineer inspects 20 lots, what is the probability that all of them will be shipped? 5. A contractor has to choose between two jobs. The first job promises a profit of $240,000 with a probability of 0.75 or a loss of $60,000 with a probability of 0.25; the second job promises a profit of $360,000 with a probability of 0.50 or a loss of $90,000 with a probability of 0.50. A. Which job should the contractor choose if he/she wants to maximize his/her expected profit? B. which job would the contractor probably choose if his/her business is in bad shape and he/she will go broke unless he/she makes a profit of at least $300,000 on the next job?

Answer 1

2) The given informations about the sets are

$\dpi{100} \fn_jvn \left | A \right |=84,\left | B \right |=99,\left | C \right |=85\\ \left | A\cap B \right |=74\\ \left | A\cap C\cap \overline{B} \right |=8\\ \left | B\cap C\cap \overline{A} \right |=9\\ \left | \overline{B}\cap \overline{C}\cap \overline{A} \right |=27$

Using DeMorgans Laws,

Now,

$\dpi{100} \fn_jvn \left | A\cap \overline{B} \right |=\left | A \right |-\left | A\cap B \right |=84-74=10\\ \left | A\cap \overline{C}\cap \overline{B} \right |=10-8=2\\ \left | \overline{C}\cap \overline{B} \right |=2+27=29\\ \left |B\cup C \right |=150-29=121\\ \left | B\cap C \right |=\left | B \right |+\left | C \right |-\left | B\cup C \right |\\ \left | B\cap C \right |=99+85-121=63\\ \left | B\cap C \cap A\right |=63-9=54\\ \left | A\cap C \right |=54+8=62\\$

A) $\dpi{100} \fn_jvn P\left ( B\cap C \cap A\right )=\frac{54}{150}={\color{Blue} 0.36}$

B)

$\dpi{100} \fn_jvn P\left ( B\cup C \cup A\right )=P(A)+P(B)+P(C)-P\left ( A\cap B \right )-P\left ( B\cap C \right )-P\left ( A\cap C \right )+-P\left ( A\cap B\cap C \right )\\ P\left ( B\cup C \cup A\right )=\frac{84+99+85-74-63-62+54}{150}={\color{Blue} 0.82}$

C)

$\dpi{100} \fn_jvn P\left ( \overline{A}\cup \overline{B} \cup C\right )=1-P\left ( \overline{\overline{A}\cup \overline{B} \cup C}\right )\\ P\left ( \overline{A}\cup \overline{B} \cup C\right )=1-P\left ( A\cap B \cap \overline{C}\right )\\ P\left ( \overline{A}\cup \overline{B} \cup C\right )=1-\frac{74-54}{150}={\color{Blue} 0.8667}$

D)

$\dpi{100} \fn_jvn P(B\cap (A\cup C))=P(B\cap A\cup B\cap C)\\ P(B\cap (A\cup C))=P(B\cap A)+P( B\cap C)-P(A\cap B\cap C)\\ P(B\cap (A\cup C))=\frac{74+121-54}{150}={\color{Blue} 0.94}$