The water supply in a town depends entirely on two pumps. A and B. The probability of pump A filling is 0.1 and the probability of pump B failing is 0.2. Calculate the probability that
(a) Both pumps are working
(b) There is no water in the town
(c) Only one pump is working
(d) There is some water in the town
Let event
A=pump A is working
B=pump B is working
Assume pumps A and B are independent.
then
A'=pump A is not working
B'=pump B is not working,
hence
P(A')=0.1 (probability that pump A fails)
P(A)=1-0.1=0.9 (probaility that pump A works)
P(B')=0.2
P(B)=1-0.2=0.8
(a) Both pumps are working
use the multiplication rule for both steps to succeed.
P(A∩B)=P(A)*P(B)
(b) Both pumps are not working
similar to (a), but both steps will fail.
P(A'∩B')
(c) Only one pump is working
P(A∩B')+P(A'∩B)
If A works, then B doesn't and vice versa.
(d) At least one pump is working:
subtract result (b) from 1, i.e. any case but two pumps failing at the same time.
(a) The probability that both pumps are working is the product of the probability of pump A filling (0.1) and the probability of pump B not failing (0.8), since these events are independent:
Probability(Both pumps are working) = 0.1 * 0.8 = 0.08
(b) The probability that there is no water in the town is the complement of the probability that there is some water in the town:
Probability(No water in the town) = 1 - Probability(Some water in the town)
(c) To calculate the probability that only one pump is working, we need to consider two cases: pump A working and pump B failing, and pump A failing and pump B working. Since these cases are mutually exclusive, we can calculate each probability separately and then sum them:
Probability(Only one pump is working) = Probability(A working, B failing) + Probability(A failing, B working)
Probability(Only one pump is working) = (0.1 * 0.2) + (0.9 * 0.8)
Probability(Only one pump is working) = 0.02 + 0.72
Probability(Only one pump is working) = 0.74
(d) The probability that there is some water in the town is the complement of the probability that there is no water in the town:
Probability(Some water in the town) = 1 - Probability(No water in the town)
To calculate the probabilities, we can use the multiplication rule for independent events.
(a) To find the probability that both pumps are working, we need to multiply the probability that pump A is filling (0.1) by the probability that pump B is not failing (1 - 0.2 = 0.8):
P(Both pumps are working) = P(A filling and B not failing) = P(A filling) * P(B not failing) = 0.1 * 0.8 = 0.08
(b) To find the probability that there is no water in the town, we need to calculate the probability that both pumps are not working:
P(No water in the town) = P(Both pumps not working) = P(A not filling and B failing) = P(A not filling) * P(B failing) = (1 - 0.1) * 0.2 = 0.18
(c) To find the probability that only one pump is working, we need to consider two cases:
- Pump A is filling, and pump B is not failing: P(A filling) * P(B not failing) = 0.1 * 0.8 = 0.08
- Pump A is not filling, and pump B is not failing: P(A not filling) * P(B not failing) = 0.9 * 0.8 = 0.72
The probability that only one pump is working is the sum of these two cases:
P(Only one pump is working) = 0.08 + 0.72 = 0.8
(d) To find the probability that there is some water in the town, we need to consider two cases:
- Both pumps are working: P(Both pumps are working) = 0.08
- Only one pump is working: P(Only one pump is working) = 0.8
The probability that there is some water in the town is the sum of these two cases:
P(There is some water in the town) = 0.08 + 0.8 = 0.88
To calculate the probabilities, we need to consider the probabilities of individual events and then use probability rules to combine them.
(a) To calculate the probability that both pumps are working, we need to find the product of the probabilities of pump A filling and pump B not failing.
P(Both pumps working) = P(Pump A fills) * P(Pump B does not fail) = 0.1 * (1 - 0.2) = 0.1 * 0.8 = 0.08
Therefore, the probability that both pumps are working is 0.08.
(b) To calculate the probability that there is no water in the town, we need to find the probability that either pump A doesn't fill or pump B fails.
P(No water in town) = P(Pump A does not fill) + P(Pump B fails) = 1 - P(Pump A fills) + P(Pump B fails) = 1 - 0.1 + 0.2 = 0.1 + 0.2 = 0.3
Therefore, the probability that there is no water in the town is 0.3.
(c) To calculate the probability that only one pump is working, we need to find the sum of the probabilities of pump A filling and pump B failing.
P(Only one pump working) = P(Pump A fills) + P(Pump B fails) = 0.1 + 0.2 = 0.3
Therefore, the probability that only one pump is working is 0.3.
(d) To calculate the probability that there is some water in the town, we need to find the complement of the probability that there is no water in the town.
P(Some water in town) = 1 - P(No water in town) = 1 - 0.3 = 0.7
Therefore, the probability that there is some water in the town is 0.7.