Probability: 'At Least' Questions
Stacey needs to guess on the last 3 questions of a quiz. Each question has 5 answer choices, one of which is correct. What is the probability that she gets at least 1 of the questions correct?
Incorrect.
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You might have gotten $$\displaystyle \frac{1}{125}$$ if you found the probability that Stacey will get all 3 correct.
$$\hspace{0.5in} \displaystyle P(\mbox{1st Correct}) \cdot P(\mbox{2nd Correct}) \cdot P(\mbox{3rd Correct}) $$
$$\hspace{0.5in} \displaystyle = \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} = \frac{1}{125}$$
Incorrect.
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It's possible to arrive at $$\displaystyle \frac{12}{125}$$ if you found the probability that Stacey will answer 2 of the questions correctly.
$$\hspace{0.5in} \displaystyle P(\mbox{1st Correct}) \cdot P(\mbox{2nd Correct}) \cdot P(\mbox{3rd Incorrect}) $$
$$\hspace{0.5in} \displaystyle = \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{4}{5} = \frac{4}{125}$$
Since there are three ways this could happen (CCW, CIC, ICC), the probability is
$$\hspace{0.5in} \displaystyle 3 \cdot \frac{4}{125} = \frac{12}{125}$$.
Incorrect.
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An answer of $$\displaystyle \frac{13}{125}$$ suggests you found the probability that Stacey will answer _more than_ 1 of the questions correctly.
Incorrect.
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You might have gotten $$\displaystyle \frac{48}{125}$$ if you calculated the probability that Stacey will answer _exactly_ 1 of the questions correctly.
$$\hspace{0.5in} \displaystyle P(\mbox{1st Correct}) \cdot P(\mbox{2nd Incorrect}) \cdot P(\mbox{3rd Incorrect}) $$
$$\hspace{0.5in} \displaystyle = \frac{1}{5} \cdot \frac{4}{5} \cdot \frac{4}{5} = \frac{16}{125}$$
Since there are three ways this could happen (CII, ICI, IIC), the probability is
$$\hspace{0.5in} \displaystyle 3 \cdot \frac{16}{125} = \frac{48}{125}$$.
The above solution method considers each way of reaching the desired event separately. Alternatively, the following shortcut can be used once you feel more comfortable with this concept:
First, you need to know that the odds that Stacey gets one of them right are 1 minus the probability that she gets all of them wrong. For each question, the probability that she gets it wrong are 4/5. The probability that she gets them all wrong are $$\frac{4}{5} \cdot \frac{4}{5} \cdot \frac{4}{5} = \frac{64}{125}$$. So the probability that she gets at least one right is $$1 - \frac{64}{125} = \frac{61}{125}$$.
Yes!
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The probability that Stacey gets a particular question wrong is $$\frac{4}{5}$$. Therefore, the probability that she gets all 3 questions wrong is
$$\hspace{0.5in} \displaystyle \frac{4}{5} \cdot \frac{4}{5} \cdot \frac{4}{5} = \frac{64}{125}$$.
By __Forbidden Combinations__, the probability that she gets at least 1 question right is
$$\hspace{0.5in} \displaystyle 1 - \frac{64}{125} = \frac{61}{125}$$.
Correct!
For Stacey to answer at least one of the three questions correctly, any of the following cases will work.
1) Stacey gets exactly one question right:
| Question 1 | Question 2 | Question 3 |
|------------|------------|------------|
| Correct | Incorrect | Incorrect |
| Incorrect | Correct | Incorrect |
| Incorrect | Incorrect | Correct |
In this case, there are three ways that it can happen, as shown in the table above. For each scenario, the probability is $$\frac{1}{5} \cdot \frac{4}{5} \cdot \frac{4}{5} = \frac{16}{125}$$. Therefore, the combined probability for Stacey to answer exactly one question correctly is $$3 \cdot \frac{16}{125} = \frac{48}{125}$$.
2) Stacey gets exactly two questions right:
| Question 1 | Question 2 | Question 3 |
|------------|------------|------------|
| Correct | Correct | Incorrect |
| Incorrect | Correct | Correct |
| Correct | Incorrect | Correct |
In this case, there are three ways that it can happen, as shown in the table above. For each scenario, the probability is $$\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{4}{5} = \frac{4}{125}$$. Therefore, the combined probability for Stacey to answer exactly two questions correctly is $$3 \cdot \frac{4}{125} = \frac{12}{125}$$.
3) Stacey gets exactly three questions right:
| Question 1 | Question 2 | Question 3 |
|------------|------------|------------|
| Correct | Correct | Correct |
In this case, there is only one way that it can happen, and the probability is $$\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} = \frac{1}{125} $$. Therefore, the combined probability for Stacey to answer exactly three questions correctly is $$1 \cdot \frac{1}{125} = \frac{1}{125} $$.
Combining the probabilities of all of these scenarios together, the total probability is $$\frac{48}{125} + \frac{12}{125}+ \frac{1}{125} = \frac{61}{125}$$.
$$\displaystyle \frac{1}{125}$$
$$\displaystyle \frac{12}{125}$$
$$\displaystyle \frac{13}{125}$$
$$\displaystyle \frac{48}{125}$$
$$\displaystyle \frac{61}{125}$$
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