Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works if and only if either 1 or 2 works; since 3 and 4 are connected in series, that subsystem works if and only if both 3 and 4 work. If components work independently of one another and P(component i works) = 0.83 for i = 1, 2 and = 0.66 for i = 3, 4,

Question

Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works if and only if either 1 or 2 works; since 3 and 4 are connected in series, that subsystem works if and only if both 3 and 4 work. If components work independently of one another and P(component i works) = 0.83 for i = 1, 2 and = 0.66 for i = 3, 4,

calculate P(system works).

Answer 1

No Picture

Answer 2

To calculate the probability that the system works, we need to consider the probabilities of the different scenarios where the system can work.

First, let's consider the scenario where both components 1 and 2 work. Since they are connected in parallel, the probability of this scenario happening is the product of their individual probabilities:

P(1 and 2 work) = P(component 1 works) * P(component 2 works) = 0.83 * 0.83 = 0.6889

Next, let's consider the scenario where components 3 and 4 work. Since they are connected in series, the probability of this scenario happening is the product of their individual probabilities:

P(3 and 4 work) = P(component 3 works) * P(component 4 works) = 0.66 * 0.66 = 0.4356

Now, since components 1 and 2 are connected in parallel, the system works if either scenario 1 or scenario 2 happens. Therefore, we need to calculate P(system works) by adding the probabilities of scenario 1 and scenario 2:

P(system works) = P(1 and 2 work) + P(3 and 4 work) - P(1 and 2 work) * P(3 and 4 work)

P(system works) = 0.6889 + 0.4356 - (0.6889 * 0.4356) = 0.6889 + 0.4356 - 0.2998

P(system works) = 0.8247

Therefore, the probability that the system works is approximately 0.8247.

Answer 3

To calculate the probability that the system works, we need to consider the probabilities of each subsystem working.

Let's start by calculating the probability that subsystem 3 and 4 work. Since they are connected in series, both components need to work for the subsystem to work. We can calculate this probability using the multiplication rule for independent events:

P(subsystem 3 and 4 work) = P(component 3 works) * P(component 4 works)
= 0.66 * 0.66
= 0.4356

Next, let's calculate the probability that either component 1 or 2 work. Since they are connected in parallel, if either component works, the subsystem will work. We can use the addition rule for mutually exclusive events:

P(subsystem 1 or 2 work) = P(component 1 works) + P(component 2 works)
= 0.83 + 0.83
= 1.66

However, the probability cannot be greater than 1, so we need to adjust this probability. Since the events are mutually exclusive, we can limit the probability to 1 using the formula:

P(subsystem 1 or 2 work) = min(P(component 1 works) + P(component 2 works), 1)
= min(0.83 + 0.83, 1)
= min(1.66, 1)
= 1

Now, we need to calculate the probability that both subsystem 3 and 4 work, and either subsystem 1 or 2 work. Since these subsystems are connected in series, both subsystems need to work for the system to work. We can calculate this probability using the multiplication rule:

P(system works) = P(subsystem 3 and 4 work) * P(subsystem 1 or 2 work)
= 0.4356 * 1
= 0.4356

Therefore, the probability that the system works is 0.4356, or 43.56%.