guessing generated answers with a generator

Question

1.) Suppose we have $m$ number of 2-choice questions. For each question, suppose the probability for its answer to be $0$ is $\lambda$ and the proability for $1$ is $1- \lambda$. Suppose each question is independent with other questions. (Thus, the distribution of the answers follow multinomial distribution.)

2.) Suppose a student knows how the answers of these questions have been genereated but just do not know the exact answers for the questions. Thus, the student generates a list of answers using the multinomial distribution with $\lambda$. What is the probability that the student got all the questions answered correctly?

Answer 1

Hint:

• What is the probability the correct answer to the first question is 0?
• What is the probability the student gives the answer 0 to the first question?
• What is the probability the correct answer is 0 and the student gives the answer 0 to the first question?
• What is the probability the correct answer is 1 and the student gives the answer 1 to the first question?
• What is the probability the student gives the correct answer to the first question?
• What is the probability the student gives the correct answer to all the questions?