How Can Probability Help Solve Complex IQ Test Problems?

[Sheldon1]

Hi to all!
I don`t know to do this tasks, so I would much apriciate if someone can help

1) The student fills IQ test (true - false based).There is m offered answers on each of N questions. For every question student knows the answer (with probability P) or answering by random (with probability 1-P), and his answer is correct with the probability of 1/m.

a) Determine the probability that the student really knew the answer to the first question, if we know that he gave a correct answer.
b) Determine the probability that the student will complete correctly the entire test.

2) Two points are randomly selected (M1 and M2) along the line L (L => 1). Determine the probability that the M1M2 <1?

Desperately need help :(

 

[mmwave]

Hello,

I am happy to help with these tasks. For the first question, we can use the conditional probability formula to determine the probability that the student really knew the answer to the first question given that they gave a correct answer. It is calculated as follows:

P(knowing the answer | correct answer) = P(knowing the answer and giving a correct answer) / P(giving a correct answer)

Since we know that the student knows the answer with a probability of P and gives a correct answer with a probability of 1/m, the numerator of this equation is P * 1/m. The denominator is simply the probability of giving a correct answer, which is equal to P * 1/m + (1-P) * 1/m = 1/m. Therefore, the final probability is P / (P + 1 - P) = P.

For part b, we can use the same logic to determine the probability of completing the entire test correctly. Since each question is independent, the probability of completing the entire test correctly is equal to the product of the probabilities of giving a correct answer for each question. This is calculated as follows:

P(completing the entire test correctly) = P(correct answer for question 1) * P(correct answer for question 2) * ... * P(correct answer for question N)

= (P * 1/m)^N = P^N / m^N

For the second question, we can use the geometric probability formula to determine the probability that M1M2 < 1. This formula is P(event) = length of event / total length. In this case, the event is M1M2 < 1, which means that the distance between the two points is less than 1. The total length is the length of the line L, which is given to be greater than or equal to 1. Therefore, the probability is P(M1M2 < 1) = 1 / L.

I hope this helps. Let me know if you have any further questions or need clarification. Best of luck with your tasks!