List of top Mathematics Questions on Probability Distribution asked in CBSE CLASS XII

It is known that 20% of the students in a school have above 90% attendance and 80% of the students are irregular. Past year results show that 80% of students who have above 90% attendance and 20% of irregular students get “A” grade in their annual examination. At the end of a year, a student is chosen at random from the school and is found to have an “A” grade. What is the probability that the student is irregular?

Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day upskilling self. The probability distribution of the number of hours spent by a student is given below:

\[ P(X = x) = \begin{cases} kx^2 & {for } x = 1, 2, 3, \\ 2kx & {for } x = 4, 5, 6, \\ 0 & {otherwise}. \end{cases} \]

Based on the above information, answer the following:

Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% of learners were self-taught using internet resources and upskilled themselves.

A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below:

\[ P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3, \\ 2kx, & \text{for } x = 4, 5, 6, \\ 0, & \text{otherwise.} \end{cases} \]

where \( x \) denotes the number of hours. Based on the above information, answer the following questions:
(i) Express the probability distribution given above in the form of a probability distribution table.
(ii) Find the value of \( k \).
(iii)(a) Find the mean number of hours spent by the student.
(iii)(b) Find \( P(1 < X < 6) \).