Answer:
a. 0.011 or 1.1%
b. 31.56% or 0.3156
c. 99.94% or 0.9994
d. 3.42% or 0.0342
Step-by-step explanation:
Given
Number of multiple choice questions = 100
Probability of success for students who have attended lectures and done their homework = 0.85
a. Using binomial distribution
Probability of correctly answering 90 or more questions out of 100
[tex]= \sum^{100}_{x=90}\left {100} \atop {x}} \right C (0.85)^x(0.15)^{100-x}\\=0.011[/tex]
Since,
In Binomial Distribution
[tex]P(X=x) =\left {h} \atop {x}} \right.C P^x q^{n-x}[/tex]
where [tex]x=0,1,...,n[/tex]
and [tex]q=1-P[/tex]
Probability is therefore 1.1% or 0.011
b. Probability of correctly answering 77 to 83 questions out of 100
[tex]= \sum^{83}_{x=77}\left {100} \atop {x}} \right C (0.85)^x(0.15)^{100-x}\\=0.3156[/tex]
The probability is therefore 31.56% or 0.3156
c. Probability of correctly answering more than 73 questions out of 100
[tex]= \sum^{100}_{x=73}\left {100} \atop {73}} \right C (0.85)^x(0.15)^{100-x}\\=0.9994[/tex]
The probability is therefore 99.94% or 0.9994
d. Assuming that the student has answered randomly
Probability of success = 1/5 = 0.2
Probability of failure = 1 - 0.2 = 0.8
Probability of answering 28 or more questions correctly
[tex]= \sum^{100}_{x=28}\left {100} \atop {x}} \right.c (0.2)^x(0.8)^{100-x}\\=0.0342[/tex]
The probability is therefore 3.42%
