Question

A quiz consists of 20 true-false questions. The score for each question is 1 point if it is answered correctly, and 0 otherwise. To get an A grade on the test, you need a total score of more than 16 points. One of the students knows the correct answer to 6 of the 20 questions. The rest she guesses at random by tossing a coin (one toss per question). What is the chance that she gets an A grade on the test?

Answer 1

less than 50:50

Answer 2

The number of questions to be answered by random guess is 14. To get an A grade the number of these questions that must be answered correctly is 11, 12, 13 or 14.
\[P(11\ correct)=\left(\begin{matrix}14 \\ 11\end{matrix}\right)0.5^{14}=0.0222\]
\[P(12\ correct)=\left(\begin{matrix}14 \\ 12\end{matrix}\right) 0.5^{14}=0.00554\]
\[P(13\ correct)=\left(\begin{matrix}14 \\ 13\end{matrix}\right)0.5^{14}=0.0008545\]
\[P(14\ correct)=\left(\begin{matrix}14 \\ 14\end{matrix}\right)0.5^{14}=0.000061\]
The chance that she gets an A grade on the test is the sum of the above probabilities:
P(A grade) = 0.0222 + 0.00554 + 0.0008545 + 0.000061 = ?