Answer to Question #326183 in Statistics and Probability for Robera

Question

Suppose that an examination consists of six true and false questions, and assume that

a student has no knowledge of the subject matter. The probability that the student will

guess the correct answer to the first question is 30%. Likewise, the probability of

guessing each of the remaining questions correctly is also 30%.

a. What is the probability of getting more than three correct answers?

b. What is the probability of getting at least two correct answers?

c. What is the probability of getting at most three correct answers?

d. What is the probability of getting less than five correct answers

Answer 1

$X-number\,\,of\,\,correct\,\,answers, X\sim Bin\left( 6,0.3 \right) \\a:\\P\left( X>3 \right) =\sum_{i=4}^6{P\left( X=i \right)}=\sum_{i=4}^6{C_{6}^{i}\cdot 0.3^i\cdot \left( 1-0.3 \right) ^{6-i}}=\\=15\cdot 0.3^4\cdot 0.7^2+6\cdot 0.3^5\cdot 0.7^1+0.3^6=0.07047\\b:\\P\left( X\geqslant 2 \right) =1-P\left( X<2 \right) =1-P\left( X=0 \right) -P\left( X=1 \right) =\\=1-0.7^6-6\cdot 0.3\cdot 0.7^5=0.579825\\c:\\P\left( X\leqslant 3 \right) =1-P\left( X>3 \right) =1-0.07047=0.92953\\d:\\P\left( X<5 \right) =1-P\left( X\geqslant 5 \right) =1-P\left( X=5 \right) -P\left( X=6 \right) =1-6\cdot 0.3^5\cdot 0.7^1-0.3^6=0.989065$