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(a) P(1 boy) P(3 boys) Then P(at least 1 boy) P(1 boy) 1 4 Another method P(at least 1 boy) (b) P(at least 1 boy and at least 1 girl) 1 1 1 P(no boy) P(no boy) 1 16 1 16 1 1 4 a b 2 1 1 16 15 16 3 8 1 4 P(2 boys) 1 16 15 16 P(3 boys) P(4 boys) 4 1 1 1 3 a ba b a b 2 1 2 4 1 3 1 1 a ba b a b 2 3 2 1 , 4 1 , 4 P(2 boys) P(4 boys) 4 1 2 1 2 a ba b a b 2 2 2 4 1 4 1 0 a ba b a b 2 4 2 3 8 1 16

P(no girl) 7 8

p p Referring to Fig. 11-28, VLN 141:4 2 = 3 115:5 V, and so 115:5 08 23:1 308 A 15=3 308

We could also have solved this problem by letting X be a random variable denoting the number of boys in families with 4 children. Then, for example, (a) becomes P(X 1) P(X 1) P(X 2) P(X 3) P(X 4) 15 16

Fig. 11-28 The line currents lag the ABC-sequence, line-to-neutral voltages by 308: IA 23:1 608 A IB 23:1 608 A IC 23:1 1808 A

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3. What functions are important to include in use case diagrams The important functions are those critical functions that convey the required functionality. 4. What criteria should be used to evaluate scenarios Every signi cant sequence of functions needs to be shown from the user s point of view. 5. What criteria should be used to evaluate state diagrams State diagrams need to show all possible transitions. Every arc in the diagram needs to have an event that caused the transition. There must also be a path from the start node to every node and from every node to a terminal node. 6. What is the advantage of a system diagram Being less formal, it can be more exible in expressing ideas about the overall system. 7. What can be a major problem with system diagrams The lack of formality can lead to ambiguous diagrams with one symbol being used for di erent ideas.

4.4. Out of 2000 families with 4 children each, how many would you expect to have (a) at least 1 boy, (b) 2 boys, (c) 1 or 2 girls, (d) no girls

Referring to Problem 4.3, we see that (a) Expected number of families with at least 1 boy (b) Expected number of families with 2 boys (c) P(1 or 2 girls) P(1 girl) P(1 boy) P(2 girls) P(2 boys) 1 4 3 8 5 8 1250 125 2000a 15 b 16 1875 3 2000a b 8 750

p The phase currents, of magnitude IPh IL = 3 13:3 A, lag the corresponding line-to-line voltages by 308: IAB 13:3 908 A IBC 13:3 308 A ICA 13:3 2108 A

2000 P(2 boys)

A three-phase, three-wire system, with an e ective line voltage 176.8 V, supplies two balanced loads, one in delta with Z 15 08 and the other in wye with ZY 10 308 . Obtain the total power.

4.5. If 20% of the bolts produced by a machine are defective, determine the probability that out of 4 bolts chosen at random, (a) 1, (b) 0, (c) less than 2, bolts will be defective.

The probability of a defective bolt is p 0.2, of a nondefective bolt is q X be the number of defective bolts. Then (a) P(X (b) P(X (c) P(X 1) 0) 2) 4 a b(0.2)1(0.8)3 1 4 a b(0.2)0(0.8)4 0 P(X 0.4096 0) P(X 0.4096 0.4096 0.4096 1) 0.8192 1 p 0.8. Let the random variable

First convert the -load to Y, and then use the single-line equivalent circuit, Fig. 11-29, to obtain the line current.

1. Draw an object model for the B&B problem (see Problem 4.3). See Fig. 8-8.

4.6. Find the probability of getting a total of 7 at least once in three tosses of a pair of fair dice.

CHAP. 11]

In a single toss of a pair of fair dice the probability of a 7 is p probability of no 7 in a single toss is q 1 p 5>6. Then P(no 7 in three tosses) and

P(at least one 7 in three tosses)

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