Applied practice on mutually exclusive events, exhaustive events, complements, independent events and conditional probability.
This video should explain all basic event problems before Bayes’ theorem.
A good solution should not start directly with numbers. First define the events, then identify the correct formula, then substitute the values.
A site engineer checks daily safety reports from a construction site. A report may contain a minor electrical fault or a plumbing fault. From previous records, \(35\%\) of the reports had a minor electrical fault and \(28\%\) had a plumbing fault. These two faults cannot occur together in the same report. Find the probability that a randomly selected report has either a minor electrical fault or a plumbing fault.
Define the events and write the correct formula first. Then compare your work with the full solution.
Since the events are mutually exclusive,
\[P(A\cup B)=P(A)+P(B)\]
\[P(A\cup B)=0.35+0.28=0.63\]
This problem is part of the BP2 basic probability video.
In an electronics lab, a circuit may have voltage fluctuation or overheating. From inspection records, \(62\%\) of circuits had voltage fluctuation, \(45\%\) had overheating, and \(20\%\) had both problems. Find the probability that a randomly selected circuit has voltage fluctuation or overheating.
Define the events and write the correct formula first. Then compare your work with the full solution.
Use the addition theorem:
\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]
\[P(A\cup B)=0.62+0.45-0.20=0.87\]
This problem is part of the BP2 basic probability video.
A factory produces sensors for measurement devices. The probability that a randomly selected sensor is defective is \(0.04\). Find the probability that the sensor is not defective.
Define the events and write the correct formula first. Then compare your work with the full solution.
Using the complement rule,
\[P(D')=1-P(D)\]
\[P(D')=1-0.04=0.96\]
This problem is part of the BP2 basic probability video.
In a computer lab, the probability that the projector fails on a given day is \(0.03\). The probability that the network cable fails on the same day is \(0.05\). Assume the two failures are independent. Find the probability that both failures occur on the same day.
Define the events and write the correct formula first. Then compare your work with the full solution.
Since the events are independent,
\[P(A\cap B)=P(A)P(B)\]
\[P(A\cap B)=0.03(0.05)=0.0015\]
This problem is part of the BP2 basic probability video.
For the same computer lab, the probability that the projector fails is \(0.03\), and the probability that the network cable fails is \(0.05\). Assume the failures are independent. Find the probability that at least one of the two failures occurs.
Define the events and write the correct formula first. Then compare your work with the full solution.
First find the probability that both occur:
\[P(A\cap B)=0.03(0.05)=0.0015\]
Now use addition theorem:
\[P(A\cup B)=0.03+0.05-0.0015=0.0785\]
This problem is part of the BP2 basic probability video.
During building inspection, \(12\%\) of walls were found to have cracks. Among all inspected walls, \(4.5\%\) had both cracks and water leakage. Find the probability that a wall has water leakage given that it has cracks.
Define the events and write the correct formula first. Then compare your work with the full solution.
Use conditional probability:
\[P(L|C)=\frac{P(L\cap C)}{P(C)}\]
\[P(L|C)=\frac{0.045}{0.12}=0.375\]
This problem is part of the BP2 basic probability video.
A programming instructor checked 200 student submissions. Out of these, 50 submissions were late. Among the late submissions, 20 were affected by server issues. Find the probability that a submission had server issues given that it was late.
Define the events and write the correct formula first. Then compare your work with the full solution.
Using conditional probability from counts:
\[P(S|L)=\frac{n(S\cap L)}{n(L)}\]
\[P(S|L)=\frac{20}{50}=0.40\]
This problem is part of the BP2 basic probability video.
A machine part is classified as having low tolerance error, medium tolerance error or high tolerance error. These three classes are mutually exclusive and exhaustive. The probability of low error is \(0.52\), and the probability of medium error is \(0.31\). Find the probability of high error.
Define the events and write the correct formula first. Then compare your work with the full solution.
Since the events are exhaustive,
\[P(L)+P(M)+P(H)=1\]
\[P(H)=1-0.52-0.31=0.17\]
This problem is part of the BP2 basic probability video.
A component failure is classified as rust failure or overload failure. These two failure modes are mutually exclusive. The probability of rust failure is \(0.18\), and the probability of overload failure is \(0.12\). Find the probability that neither failure occurs.
Define the events and write the correct formula first. Then compare your work with the full solution.
Since the events are mutually exclusive,
\[P(R\cup O)=0.18+0.12=0.30\]
Therefore,
\[P(\text{neither})=1-P(R\cup O)=1-0.30=0.70\]
This problem is part of the BP2 basic probability video.
The probability of rain on a day is \(0.30\). The probability of a small internet outage is \(0.10\). Assume rain and internet outage are independent. Find the probability that there is an internet outage given that it rains.
Define the events and write the correct formula first. Then compare your work with the full solution.
For independent events,
\[P(O|R)=P(O)\]
\[P(O|R)=0.10\]
This problem is part of the BP2 basic probability video.
In a road survey, \(25\%\) of roads are old. Among old roads, \(40\%\) need urgent repair. Find the probability that a randomly selected road is old and needs urgent repair.
Define the events and write the correct formula first. Then compare your work with the full solution.
Use multiplication theorem:
\[P(O\cap R)=P(O)P(R|O)\]
\[P(O\cap R)=0.25(0.40)=0.10\]
This problem is part of the BP2 basic probability video.
A die is thrown once. Let \(A\) be the event of getting an even number and \(B\) be the event of getting a number greater than 4. Find \(P(A)\), \(P(B)\), \(P(A\cap B)\), and \(P(A\cup B)\).
Define the events and write the correct formula first. Then compare your work with the full solution.
\[P(A)=\frac{3}{6}=\frac{1}{2}\]
\[P(B)=\frac{2}{6}=\frac{1}{3}\]
\[A\cap B=\{6\},\quad P(A\cap B)=\frac{1}{6}\]
\[P(A\cup B)=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{2}{3}\]
This problem is part of the BP2 basic probability video.