Answer to Question #231906 in Statistics and Probability for yes

Question

A math competition hosted by HOD has the following rules. Each team will be composed of four (4) people and each

team will try to answer ten (10) math questions of the same difficulty. You are only required to answer seven (7) out

of the 10 questions correctly. If you are able to answer 7 questions, the team should raise their flag and a proctor will

approach them to verify if they successfully answered 7 questions correctly.

If in case the correct answer is less than 7, the proctor will give you another chance to work on achieving 7 correct

answers. If the team already got 7 correct answers, the proctor will record that time. The team with the best time will

win 1% of the PHP 67,000,000,000 HOD has in their funds.

The probability of answering a question correctly is 0.45.

Let X be the number of getting correct answers out of the 10 questions.

Let Y be the number of times the proctors approach the team to verify if the team successfully answered 7 questions

correctly.

Answer 1

P(Solving a question correctly) = 0.45

Number of questions = 10

X is a random variable denoting the number of correct questions

Therefore 

$X—Binomial(n = 10,p = 0.45) \\ P(X = x)= \begin{pmatrix} 10 \\ x \end{pmatrix} (0.45)^x (1-0.45)^{ 10-x}, x = 0,1,2, ...,10$

P(Solving seven or more questions correct) $= P(X≥ 7)$

$= P(X = 7)+ P(X = 8)+ P(X = 9)+ P(X = 10)$

$=\begin{pmatrix} 10 \\ 7 \end{pmatrix} (0.45)^7(0.55)^{10-7}+\begin{pmatrix} 10 \\ 8 \end{pmatrix} (0.45)^8(0.55)^{10-8} + \begin{pmatrix} 10 \\ 9 \end{pmatrix} (0.45)^9(0.55)^{10-9}+\begin{pmatrix} 10 \\ 10 \end{pmatrix} (0.45)^{10}(0.55)^{10-10}\\$

$= 0.07460311 + 0.02288959 + 0.004161744 + 0.0003405063 = 0.101995$

P(Solving seven or more questions correct) $= 0.102$

Thus,

Y follows a Geometric distribution with $p = 0.102$

$P(Y = y) = 0.102 * (1- 0.102)^{y-1}, y= 1,2,3,...\\ P(Y = y) = 0.102 * (1- 0.102)^{y-1}, y= 1,2,3,...\\$

P(Proctor checking 5 times until I solved 7 question correctly)$= P(Y = 5)$

$P(Y = 5) = 0.102 * (1— 0.102)5-1 = 0.0663$

P(Proctor checking 5 times until I solved 7 question correctly) = 0.0663