Let X denote the number of hours you play during a randomly selected day. The probability that X can take values x has the following form, where c is some constant: \[ P(X = x) = \begin{cases} 0.1, & \text{if } x = 0 \\ cx, & \text{if } x = 1 \text{ or } x = 2 \\ c(5 - x), & \text{if } x = 3 \text{ or } x = 4 \\ 0, & \text{otherwise} \end{cases} \] Match List-I with List-II: Choose the correct answer from the options given below: List-I List-II (A) c (I) 0.75 (B) P(X ≤ 2) (II) 0.3 (C) P(X = 2) (III) 0.55 (D) P(X ≥ 2) (IV) 0.15
(A) - (IV), (B) - (III), (C) - (II), (D) - (I)
To solve this problem, we will determine the constant \( c \) and calculate the probabilities as needed.
- The probability \( P(X = x) \) is given by:
- \( P(X = 0) = 0.1 \)
- \( P(X = 1) = c \times 1 = c \)
- \( P(X = 2) = c \times 2 = 2c \)
- \( P(X = 3) = c(5 - 3) = 2c \)
- \( P(X = 4) = c(5 - 4) = c \)
- According to the properties of probability, \( \sum P(X = x) = 1 \). Therefore, we calculate:
- Solving for \( c \):
- Next, calculate \( P(X \leq 2) \):
- Calculate \( P(X = 2) \):
- Calculate \( P(X \geq 2) \):
\((A) - (IV), (B) - (III), (C) - (II), (D) - (I)\) is correct .
Important Questions from Probability
A die is rolled thrice. What is the probability of getting a number greater than 4 in the first and the second throws, and a number less than 4 in the third throw?
Two dice are thrown simultaneously. If \( X \) denotes the number of fours, then the expectation of \( X \) will be:
If the random variable \( X \) has the following distribution:
X 0 1 2 otherwise P(X) k 2k 3k 0 Match List-I with List-II:
List-I List-II (A) k (I) \(\frac{5}{6}\) (B) P(X < 2) (II) \(\frac{4}{3}\) (C) E(X) (III) \(\frac{1}{2}\) (D) P(1 ≤ X ≤ 2) (IV) \(\frac{1}{6}\) Choose the correct answer from the options given below:
For the differential equation \( (x \log_e x) dy = (\log_e x - y) dx \):
(A) Degree of the given differential equation is 1.
(B) It is a homogeneous differential equation.
(C) Solution is \( 2y \log_e x + A = (\log_e x)^2 \), where A is an arbitrary constant.
(D) Solution is \( 2y \log_e x + A = \log_e (\log_e x) \), where A is an arbitrary constant.
Choose the correct answer from the options given below:
Which of the following is the probability of \( x \) successes in a binomial distribution with number of trials \( n \) and probability of success as \( \theta \) ( \( 0 < \theta < 1 \) ) in each trial?
