Step-by-step explanation:
Here, the total number of T/F question = 10
The minimum answers needed correctly answered = 50%
Now, 50% of 10 = 5 questions
So, student needs to answer at least 5 questions correctly.
Here, the possibility of answering a question correctly = [tex]\frac{1}{2}[/tex] = p = 0.5
Also, the possibility of answering a question wrong = [tex]\frac{1}{2}[/tex] = q = 0.5
Now, to pass he needs to answer 5 or more ( = 5, 6 , 7 , 8 , 9 or 10) answers correctly.
P(answering 5 correct answer) = [tex]^{10}C_5(0.5)^5(0.5)^5 =252 (0.5)^{10} = 0.246[/tex]
P(answering 6 correct answer) = [tex]^{10}C_6(0.5)^6(0.5)^4 =210 (0.5)^{10} = 0.2050[/tex]
P(answering 7 correct answer) = [tex]^{10}C_7(0.5)^7(0.5)^3 =120 (0.5)^{10} = 0.1171[/tex]
P(answering 8 correct answer) = [tex]^{10}C_8(0.5)^8(0.5)^2 =120 (0.5)^{10} = 0.0439[/tex]
P(answering 9 correct answer) = [tex]^{10}C_9(0.5)^9(0.5)^1 = 0.0097[/tex]
P(answering 10 correct answer) = [tex]^{10}C_{10}(0.5)^7(0.5)^3 =1 (0.5)^{10} = 0.00097[/tex]
So, the total Probability
= (0.246) + (0.2050)+ (0.1171) +(0.0439) + (0.0097) + (0.00097)
= 0.62267 ≈ 62.2 %
Hence, the probability that the student passes the quiz with a grade of at least 50% of the questions correct is 0.62267.
