Envelopes come manufactured with glue axed at three separate locations on the ap

Question

Envelopes come manufactured with glue axed at three separate locations on the ap. After wetting the glue, all that is required to have the envelope seal is for two of the glue spots to stick to the envelope. The automatic envelope stung and sealing machine in the oce will insert a letter into one of these envelopes and then wet the three glue spots. The amount of water applied to each glue spot follows a normal distribution with mean of 10 ml and standard deviation of 1.5 ml. If less than 8.5 ml of water is applied the glue will not activate (no adherence). If more than 13 ml of water is applied the glue will wash away (no adherence).
Part 1a: What is the probability that an envelope will seal?

Part 1b: On average, what percentage of envelopes fail to seal?

Explanation / Answer

Ans:

mean=10

standard deviation=1.5

a)P(envelope will seal)=P(<8.5<x<13)

z(8.5)=(8.5-10)/1.5=-1

z(13)=(13-10)/1.5=2

P(-1<z<2)=P(z<2)-P(z<-1)

=0.9772-0.1587=0.8186

b)

P(envelop will fail)=P(x<8.5 or x>13)=P(z<-1 or z>2)

=P(z<-1)+P(z>2)=0.1587+0.0228=0.1814

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