At least one hose is in use at both islands with probability 0.80.
(a) What is \(P(X=1\) and \(Y-1)\) ?
To calculate the probability of two events happening together, we multiply the probabilities of each event happening individually. Since the events are independent, we don't need to worry about the order in which they happen.
So, to calculate \(P(X=1\) and \(Y=1)\), we multiply the probability that X is equal to 1 and the probability that Y is equal to 1. The probability that X is equal to 1 is 0.20 (from the table). The probability that Y is equal to 1 is 0.07 (from the table). Therefore, the probability that X is equal to 1 and Y is equal to 1 is 0.20 * 0.07 = 0.014.
Answer: 0.014
(b) Compute \(P(X\le1~and~Y\le1)\) This can also be written as: $P(X\le1~and~Y\le1)=P(X=0~or~X=1)\times P(Y=0~or~Y=1)$
To calculate the probability of one of two events happening, we add the probabilities of each event happening individually.
So, to calculate \(P(X\le1~and~Y\le1)\), we add the probability that X is equal to 0 and the probability that X is equal to 1. We then multiply this result by the probability that Y is equal to 0 or the probability that Y is equal to 1.
The probability that X is equal to 0 is 0.10 (from the table). The probability that X is equal to 1 is 0.20 (from the table). The probability that Y is equal to 0 is 0.10 (from the table). The probability that Y is equal to 1 is 0.07 (from the table).
Therefore, the probability that X is equal to 0 or equal to 1 and Y is equal to 0 or equal to 1 is 0.10 + 0.20 * 0.10 + 0.20 * 0.07 = 0.34.
Answer: 0.34
(c) Give a word description of the event { X≠0 and \(Y\\e0;\) }
The event { X≠0 and \(Y\\e0;\) } is the event that at least one hose is in use at both islands.
Another way to describe this event is that it is the event that both hoses are not in use at either island.
Compute the probability of this event.
The probability of this event is 1 - the probability that both hoses are in use at either island.
The probability that both hoses are in use at the self-service island is 0.06 (from the table). The probability that both hoses are in use at the full-service island is 0.14 (from the table).
Therefore, the probability that both hoses are in use at either island is 0.06 + 0.14 = 0.20.
Therefore, the probability that at least one hose is in use at both islands is 1 - 0.20 = 0.80.
Answer: 0.80
Overall, the image shows the joint probability table for the number of hoses in use at a self-service and a full-service island at a particular time. The table can be used to answer various questions about the probability of different events happening, such as the probability that at least one hose is in use at both islands.