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# solution15 - Probability for Engineering Applications...

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Probability for Engineering Applications SOLUTION Assignment #15 The Central Limit Theorem has infinite applications. Here are four short examples. 1. (2 pts) A fair coin is tossed 1000 times. Estimate the probability that the number of heads is between 470 and 530. The number of independent trials is 250 , 500 , 1000 2 = = = = = npq np m n s . 9428 . 0 0286 . 0 2 1 ) 90 . 1 ( 2 1 ) 90 . 1 ( ) 90 . 1 ( 250 500 530 250 500 470 530 470 [ = × - = - = - - = - < - < - = < < Q Q Q m N P N P s 2. (2 pts) A fair die is tossed 100 times. Estimate the probability that the total number of spots is between 330 and 380. (First compute the expected number of spots and its variance for a random toss of the die.) (cf. 4.24) Let i X be the number of spots on the ith trial 92 . 2 ]) [ ( ] [ ] [ 167 . 15 6 36 25 16 9 4 1 ] [ 5 . 3 6 6 5 4 3 2 1 ] [ 2 2 2 = - = = + + + + + = = + + + + + = i i i i i X E X E X VAR X E X E For 100 tosses, 292 ] [ 100 ] [ 350 ] [ 100 ] [ 100 100 = = = = i i X VAR S VAR X E S E 8503 . 0 0357 . 114 . 0 1 ) 76 . 1 ( ) 17 . 1 ( 1 ) 76 . 1 ( ) 17 . 1 ( 292 350 380 292 350 292 350 330 ] 380 330 [ 100 100 = - - = - - = - - = - < - < - = < < Q Q Q Q S P S P

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3. (3pts) The cartridge of a heavily used printer has an exponential lifetime with mean 4 weeks. Use the Central Limit Theorem to find out how many cartridges should be bought at the beginning of a year so that the probability of not running out during the year is greater than 0.99 (cf. 4.26) Let X i be the lifetime of the
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solution15 - Probability for Engineering Applications...

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