pls help pls help probability

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.

What is the probability that the proctor checked your answers 5 times until you successfully answered 7 correct questions?

Answer 1

There are 10 questions.

Probability of answering 7, 8 , 9 , 10 correctly.

Binomial distribution, p = 0.45, q = 1 - 0.45 = 0.55

p (7 correct) = C (10,7) p⁷ q ³

p (8 correct) = C (10,8) p⁸ q²

p (9 correct) = C (10,9) p⁹ q

p (10 correct) = p¹⁰

Add these four up, giving probability of answering 7,8,9,10 correctly. Let's call this number z.

If the proctor has to check 5 times, it means the first four times you have less than 7, and the 5th time you have 7 or more.

Probability of this is (1-z)⁴ z

You can find this.