An automated filling operation, the probability of an incorrect fill when the process is operated at a low speed is 0.001. when the process is operated at a high speed, the probability of an incorrect fill is 0.01. assume that 20% of the containers are filled when the process is operated at a high speed and the remainder are filled when the process is operated at a low speed. round your answers to four decimal places (e. g. 98.7654). (a) what is the probability of an incorrectly filled container? (b) if an incorrectly filled container is found, what is the probability that it was filled during the high-speed operation?

(a) 0.0028 (b) 0.7143

Step-by-step explanation:

First of all, let's define the following events:

I: a container is filled incorrectly

L: a process is operated at a low speed

H: a process is operated at a high speed

S: the sample space

Then, we have according to the original text that

P(I | L) = 0.001, P(I | H) = 0.01, P(H) = 0.2, P(L) = 0.8

and we are seeking the following probabilities (a) P(I) and (b) P(H | I)

(a) P(I) = P(I∩S) = P[I∩(L∪H)] (because L∪H=S)

          = P[(I∩L)∪(I∩H)] (distributive laws)

          = P(I∩L)+P(I∩H) (because L and H are mutually exclusive)

          = P(I | L)P(L)+P(I | H)P(H)

          = (0.001)(0.8)+(0.01)(0.2) = 0.0028

On the other hand,

(b) P(H | I) =

               =

               =

               = 0.7143

"it's not the heat, it's the humidity".   that's a partly valid phrase you may have heard in the summer, but it's actually both.   the heat index, also known as the apparent temperature, is what the temperature feels like to the human body when relative humidity is combined with the air temperature.   this has important considerations for the human body's comfort.   when the body gets too hot, it begins to perspire or sweat to cool itself off.   if the perspiration is not able to evaporate, the body cannot regulate its temperature.   evaporation is a cooling process.   when perspiration is evaporated off the body, it effectively reduces the body's temperature.   when the atmospheric moisture content (i.e. relative humidity) is high, the rate of perspiration from the body decreases.   in other words, the human body feels warmer in humid conditions.   the opposite is true when the relative humidity decreases because the rate of perspiration increases.   the body actually feels cooler in arid conditions.   there is direct relationship between the air temperature and relative humidity and the heat index, meaning as the air temperature and relative humidity increase (decrease), the heat index increases (decreases).

step-by-step explanation: