A multiple choice question has $4$ options, of which only one is correct. If a student answers the question correctly, $2$ marks will be awarded. If a student answers the question wrongly, $1$ mark will be deducted. No marks will be awarded or deducted for leaving the question blank. Suppose Anthony does not know the answer to the question. Should he choose the answer randomly or leave the question blank? Explain your answer.

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Grade 7

Percent of a number: tax, discount, and more

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Q. A multiple choice question has

4

options, of which only one is correct. If a student answers the question correctly,

2

marks will be awarded. If a student answers the question wrongly,

1

mark will be deducted. No marks will be awarded or deducted for leaving the question blank. Suppose Anthony does not know the answer to the question. Should he choose the answer randomly or leave the question blank? Explain your answer.

Calculate Expected Value Randomly: Calculate the expected value if Anthony chooses an answer randomly. $\newline$ There are $4$ options, and only one is correct. The probability of choosing the correct answer is $\frac{1}{4}$ , and the probability of choosing a wrong answer is $\frac{3}{4}$ . $\newline$ Expected value for choosing randomly = (Probability of correct answer $\times$ Marks for correct answer) + (Probability of wrong answer $\times$ Marks for wrong answer) $\newline$ = $(\frac{1}{4} \times 2) + (\frac{3}{4} \times -1)$ $\newline$ = $0.5 - 0.75$ $\newline$ = $-0.25$
Calculate Expected Value Leaving Blank: Calculate the expected value if Anthony leaves the question blank. $\newline$ If Anthony leaves the question blank, he neither gains nor loses any marks. $\newline$ Expected value for leaving blank = $0$ marks
Compare Expected Values: Compare the expected values to make a decision. $\newline$ The expected value for choosing randomly is $-0.25$ marks, and the expected value for leaving the question blank is $0$ marks. Since $-0.25$ is less than $0$ , it is statistically better for Anthony to leave the question blank.