Question

A student takes a ten-question true-false quiz, but did not study and randomly guesses each answer. find the probability that the student passes the quiz with a grade of at least 40% of the questions correct. (round your answer to three decimal places.)

Answer 1

Number of questions = 10
Probability of getting correct = 0.5 (because it is a true - false answer)
40% of 10 question = 4 questions

Find P( ≥ 4 questions correct ):

$\text {Probability of getting 4 correct = }\left(\begin{array}{c}10\\4\end{array}\right) \bigg( \dfrac{1}{2} \bigg)^{10} = \dfrac{210}{1024}$

$\text {Probability of getting 5 correct = }\left(\begin{array}{c}10\\5\end{array}\right) \bigg( \dfrac{1}{2} \bigg)^{10} = \dfrac{252}{1024}$

$\text {Probability of getting 6 correct = }\left(\begin{array}{c}10\\6\end{array}\right) \bigg( \dfrac{1}{2} \bigg)^{10} = \dfrac{210}{1024}$

$\text {Probability of getting 7 correct = }\left(\begin{array}{c}10\\7\end{array}\right) \bigg( \dfrac{1}{2} \bigg)^{10} = \dfrac{120}{1024}$

$\text {Probability of getting 8 correct = }\left(\begin{array}{c}10\\8\end{array}\right) \bigg( \dfrac{1}{2} \bigg)^{10} = \dfrac{45}{1024}$

$\text {Probability of getting 9 correct = }\left(\begin{array}{c}10\\9\end{array}\right) \bigg( \dfrac{1}{2} \bigg)^{10} = \dfrac{10}{1024}$

$\text {Probability of getting 10 correct = }\left(\begin{array} {c}10\\10\end{array}\right) \bigg( \dfrac{1}{2} \bigg)^{10} = \dfrac{1}{1024}$

P(≥ 4 questions correct):

$\dfrac{210}{1024} + \dfrac{252}{1024} + \dfrac{210}{1024} + \dfrac{120}{1024} + \dfrac{45}{1024} + \dfrac{10}{1024} + \dfrac{1}{1024} = \dfrac{848}{1024} = 0.828$ (3 d.p.)

Answer:P(≥ 40%) =  0.828