Quizz question
Task number: 4506
In a quizz, students are asked to decide whether a statement about the probabilities of phenomena when rolling a die is true. The probability that they answer correctly is \(\frac23\). The first two students Paul and Simon in the quizz answered the same.
Is it certain that they answered correctly? If not, what is the probability that they answered right?
Solution
Probability space has 4 elementary events: The event \(A\), where both are right with probability \(\frac23\cdot\frac23=\frac49\), Paul is right and Simon is wrong with probability \(\frac29\), with equal probability it comes out that Paul is wrong but Simon is right, and finally it can happen that neither is right and this event has probability \(\frac19\).
The event \(B\) that both come up with the same answer, has \(P(B)=\frac49+\frac19=\frac59\).
Then \(P(A|B)=\frac{4/9}{5/9}=\frac45\).
Answer
The probability of both having the correct result is 80 %.Variant
Would the probability change if the quizz question had five possible answers, with students again answering it correctly with probability \(\frac23\) and all four wrong answers having the same probability of choice?Solution
Wrong answers have probability \(\frac13 : 4 =\frac{1}{12}\).
The probability that both students choose the first wrong answer is \(\frac{1}{12}\cdot\frac{1}{12}=\frac{1}{144}\) and that both students equally choose any of the four wrong answers is \(\frac{4}{144}=\frac{1}{36}\).
Now, the probability of the event \(B\) that they both come up with the same answer, is \(P(B)=\frac49+\frac1{36}=\frac{17}{36}\).
Then \(P(A|B)=\frac{4/9}{17/36}=\frac{16}{17}\doteq 0{,}941\).
Answer
The probability increases to about 94 %.Comment
The probability space can be illustrated as a unit square, where the probability of an event corresponds to the area of the corresponding pattern and the conditional probability to the proportion of areas.
