Precision components are made by machines A, B and C. Machines A and C each make \(30 \%\) of the components with machine \(\mathrm{B}\) making the rest. The probability that a component is acceptable is \(0.91\) when made by machine A, \(0.95\) when made by machine B and \(0.88\) when made by machine \(\mathrm{C}\). (a) Calculate the probability that a component selected at random is acceptable. (b) A batch of 2000 components is examined. Calculate the number of components you expect are not acceptable.

Understanding the total probability theorem is crucial when dealing with complex systems where an event can occur through several different paths. In our case, a precision component being acceptable can happen through its production by either machine A, B, or C.

Essentially, the theorem allows us to calculate the probability of a desired event, 'Acceptable', as the sum of the probabilities of 'Acceptable' given each production path, weighted by the probability of each path. It's like calculating the overall chance of rain by considering the chance of rain in different parts of a city and how much those parts contribute to the city's area.

For a clearer vision, think of it this way: If you have three different doors to enter a building and each door is of varying size, the total probability of entering the building is the sum of the probabilities of entering through each door, taking into account their sizes. This is exactly what we do with the total probability theorem; for each machine (a 'door'), we multiply the chance it makes an acceptable component (the 'probability of entering') by how much it is used (the 'size of the door').

The complement rule is a simple yet fundamental concept in probability theory. It reflects the idea that the probability of an event not occurring is simply one minus the probability of the event occurring. In essence, it's used to find the 'other side of the coin.'If an event is symbolized by 'A', then 'not A', or the complement, is everything that isn't 'A'. Think of it as a pizza: if 'A' represents the slices with pepperoni, then 'not A' represents all the slices without pepperoni. The complement rule tells us that the chance of getting a non-pepperoni slice is just one minus the chance of getting a pepperoni slice.

Applying this to our exercise, if we know the probability of picking an acceptable component, then the probability of picking a not acceptable one is the 'rest of the chances.' It's a useful shortcut to avoid complex calculations for the complementary event when we already know the main event's probability.

The concept of expected value is a cornerstone in probability and statistics, informing us about the predicted average outcome of an event if it were to be repeated many times. It's the probabilistic equivalent of an average, but with weights assigned to each outcome based on how likely they are to occur.Imagine you're playing a game where you roll a die, and you earn dollars equal to the number shown on the die. You might not earn six dollars every roll, but over many rolls, you'd average out to some consistent number. That number is the expected value. In our example, we use the expected value to predict the number of not acceptable components by multiplying the total number of components by the probability that any one component is not acceptable. It's like expecting a certain number of rainy days in a month based on weather statistics. It doesn't mean it will rain exactly that many days, but over several months, the average should be close to our expected figure.