What’s the probability of rolling a die three times in a row and getting a six each time? We can calculate the probability because of the rule of independence. The probability of two or more independent events occurring is found by multiplying the probability of each event occurring.
1.
Janice is an electrician. She always worries that her outlet
tester is faulty, so out of a large abundance of caution, she uses a
second outlet tester from a different brand to double check every
outlet. If the probability of a faulty outlet tester is 0.001 from both
brands, and if we are assured the brands’ functionality is independent
of one another, what is the probability that both outlet testers give an
incorrect reading?
It is safe to assume that the first outlet tester’s faultiness does not impact the chance of the second outlet tester being faulty, as they are different brands. That means we can find the probability of the event of both testers being faulty by multiplying these two probabilities:
\[0.001 \cdot 0.001 = 0.000001\]
There is a 0.00001% chance that both outlet testers are "faulty".
2.
An online plagiarism detection tool outputs a percentage chance
that a student’s submission was generated by AI. A professor checks in
on three submissions made by a student. The plagiarism detection tool
says the three submissions are 85%, 90%, and 95% likely to be AI
generated.
What’s the probability that none of the three submissions are written by AI? Assume that each percentage generated by the detector is independent of one another.
If the first submission is 85% likely AI, then there is a 15% chance that the submission was not AI generated. Similarly, there is a 10% and 5% chance that the second and third papers are not AI generated.
The probability of three independent events occurring is calculated by multiplying the probability of each event occurring.
\[0.15 \cdot 0.10 \cdot 0.05 = 0.00075\]
There is a 0.075% chance that this student did not submit an AI-generated submission. This indicates that the student likely cheated on one or more submissions.
3.
Friends are trying to play a game of soccer outside either Friday,
Saturday, or Sunday. The weather forecast estimates the probability of
rain for Friday to be 0.3, for Saturday to be 0.4, and for Sunday to be
0.5. Assuming the event of rain is independent for each day (which, in
reality, the events are likely actually not independent), what is the
probability that all three days will rain?
If each event is independent, the probability of all three events is
\[0.3 \cdot 0.4 \cdot 0.5 = 0.06\]
so there is a 6% chance that all days will have rain.
1. A student has three questions left on a test that they do not know how to answer. They estimate that they need to get at least one question right in order to pass the test. If the questions are multiple choice with five options, what is the probability that they get all three questions wrong? Assume that the event of getting one question correct is independent of the event of getting another question correct.
2. What is the probability that a family has four children that are all boys? Assume the event of having a boy is independent of the event of having another boy.
3. An event is planned for one of three days. Rain would cancel the event. Assume that the event of rain on one day is independent of the event of rain on another day. If day one has a probability of 0.5 of rain, day two has a probability of 0.6 of rain, and day three has a probability of 0.2 of rain, what is the probability that all three days will have rain?
1. Think of some events that would be dependent, meaning the occurrence of one event changes the probability of the occurrence of another event. One example would be the event that it rains on a certain day and the probability that an outside baseball game is canceled on a certain day. If it rains, the probability of the game getting canceled increases.
We cannot use this multiplication idea for probabilities if two events are dependent.