By means of extensive testing, a manufacturer has determined that its Super Sucker Vacuum Cleaner models have an expected life that is exponential with a mean of four years. Find the probability that one of these cleaners will have a life that ends

a. After the initial four years of service.

b. Before four years of service are completed.

c. Not before six years of service.

Question

By means of extensive testing, a manufacturer has determined that its Super Sucker Vacuum Cleaner models have an expected life that is exponential with a mean of four years. Find the probability that one of these cleaners will have a life that ends

a. After the initial four years of service.

b. Before four years of service are completed.

c. Not before six years of service.

a. After the initial four years of service.

b. Before four years of service are completed.

c. Not before six years of service.

Accepted Answer

MTBF = 4 years

a. [latex]T = 4[/latex] years:

[latex]T/MTBF = \frac{4 years}{4 years}= 1.0[/latex]

From Table 4S.1 [latex]e^{- 1.0} = .3679[/latex].

Table 4S.1 Values of [latex]e^{-T/MTBF}[/latex]

T/MTBF	[latex]e^{-T/MTBF}[/latex]	T/MTBF	[latex]e^{-T/MTBF}[/latex]	T/MTBF	[latex]e^{-T/MTBF}[/latex]
0.10	.9048	2.60	.0743	5.10	.0061
0.20	.8187	2.70	.0672	5.20	.0055
0.30	.7408	2.80	.0608	5.30	.0050
0.40	.6703	2.90	.0550	5.40	.0045
0.50	.6065	3.00	.0498	5.50	.0041
0.60	0.5488	3.10	.0450	5.60	.0037
0.70	.4966	3.20	.0408	5.70	.0033
0.80	.4493	3.30	.0369	5.80	.0030
0.90	.4066	3.40	.0334	5.90	.0027
1·00	.3679	3.50	.0302	6·00	.0025
1.10	.3329	3.60	.0273	6.10	.0022
1.20	.3012	3.70	.0247	6.20	.0020
1.30	.2725	3.80	.0224	6.30	.0018
1.40	.2466	3.90	.0202	6.40	.0017
1.50	.2231	4·00	.0183	6.50	.0015
1.60	.2019	4.10	.0166	6.60	.0014
1.70	.1827	4.20	.015	6.70	.0012
1.80	.1653	4.30	.0136	6.80	.0011
1.90	.1496	4.40	.0123	6.90	.0010
2·00	.1353	4.50	.0111	7·00	.0009
2.10	.1255	4.60	.0101
2.20	.1108	4.70	.0091
2.30	.1003	4.80	.0082
2.40	.00907	4.90	.0074
2.50	.00821	5·00	.0067

b. The probability of failure before [latex]T = 4[/latex] years is [latex]1 - e ^{- 1} , or 1 - .3679 = .6321[/latex].

c. [latex]T = 6[/latex] years:

[latex]T/MTBF = \frac{4 years}{4 years}= 1.50 [/latex]

From Table 4S.1 , [latex]e ^{-1.5} = .2231 [/latex].

Question 4S.2: By means of extensive testing, a manufacturer has determined...

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