taiwan-gsat 2021 Q7

taiwan-gsat · Other · gsat__math 5 marks Conditional Probability Confusion Matrix / Misidentification Probability Table
A psychologist conducted an experiment with 1000 subjects in a dark room, where each subject had to observe and identify three digit cards: 6, 8, and 9. The probability of mistaking the actual digit for another digit is shown in the following table:
\backslashbox{Actual Digit}{Seen as}689Other
60.40.30.20.1
80.30.40.10.2
90.20.20.50.1

For example: The actual digit 6 is seen as 6, 8, 9 with probabilities 0.4, 0.3, 0.2 respectively, and is seen as another digit with probability 0.1. Based on the above experimental results, select the correct options.
(1) If the actual digit is 8, then there is at least a 50\% chance it will be seen as 8
(2) If the actual digit is 6, then there is a 60\% chance it will be seen as not 6
(3) Among the three digits 6, 8, 9, the digit 9 has the lowest probability of being misidentified
(4) If the digit seen is 6, then the probability that it is actually 6 is less than 50\%
(5) If the digit seen is 9, then the probability that it is actually 9 is greater than $\frac { 2 } { 3 }$
A psychologist conducted an experiment with 1000 subjects in a dark room, where each subject had to observe and identify three digit cards: 6, 8, and 9. The probability of mistaking the actual digit for another digit is shown in the following table:

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\backslashbox{Actual Digit}{Seen as} & 6 & 8 & 9 & Other \\
\hline
6 & 0.4 & 0.3 & 0.2 & 0.1 \\
\hline
8 & 0.3 & 0.4 & 0.1 & 0.2 \\
\hline
9 & 0.2 & 0.2 & 0.5 & 0.1 \\
\hline
\end{tabular}
\end{center}

For example: The actual digit 6 is seen as 6, 8, 9 with probabilities 0.4, 0.3, 0.2 respectively, and is seen as another digit with probability 0.1. Based on the above experimental results, select the correct options.\\
(1) If the actual digit is 8, then there is at least a 50\% chance it will be seen as 8\\
(2) If the actual digit is 6, then there is a 60\% chance it will be seen as not 6\\
(3) Among the three digits 6, 8, 9, the digit 9 has the lowest probability of being misidentified\\
(4) If the digit seen is 6, then the probability that it is actually 6 is less than 50\%\\
(5) If the digit seen is 9, then the probability that it is actually 9 is greater than $\frac { 2 } { 3 }$