Final answer:
To calculate the probability of a JEE (Mains) score over 200 with a mean of 180 and variance of 400, first convert the score to a Z-score and then find the corresponding probability using the standard normal distribution.
Step-by-step explanation:
To find the probability that a randomly selected student has a score over 200 on the JEE (Mains), which is normally distributed with a mean score of 180 and a variance of 400, we first need to convert the score of 200 to a Z-score. The Z-score can be found using the formula Z = (X - μ) / σ, where X is the score of interest, μ is the mean, and σ is the standard deviation. In this case, the standard deviation is the square root of the variance, which is √400 = 20. The Z-score for a score of 200 is therefore (200 - 180) / 20 = 1.
Next, we consult the standard normal distribution table or use a statistical software to find the probability that Z is greater than 1. This gives us the probability that a student scores above 200. Since the standard normal distribution table gives the area to the left of the Z-score, we subtract this value from 1 to find the area to the right, which represents the probability of scoring more than 200.