Let $ x $ and $ y $ be random variables, not necessarily independent, that take real values in the interval $[0,1]$. Let $ z = xy $ and let the mean values of $ x, y, z $ be $ \bar{x} , \bar{y} , \bar{z} $, respectively. Which one of the following statements is TRUE?
A bag contains 10 red balls and 15 blue balls. Two balls are drawn randomly without replacement. Given that the first ball drawn is red, the probability (rounded off to 3 decimal places) that both balls drawn are red is ________
Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEAD on the first throw, and C be the event that denotes HEAD on the second throw. Which of the following statements is/are TRUE?
In an examination, a student can choose the order in which two questions (QuesA and QuesB) must be attempted.
- If the first question is answered wrong, the student gets zero marks.
- If the first question is answered correctly and the second question is not answered correctly, the student gets the marks only for the first question.
- If both the questions are answered correctly, the student gets the sum of the marks of the two questions.
The following table shows the probability of correctly answering a question and the marks of the question respectively.
| question | Probability of answering correctly | marks |
| QuesA | 0.8 | 10 |
| QuesB | 0.5 | 20 |
Assuming that the student always wants to maximize her expected marks in the examination, in which order should she attempt the questions and what is the expected marks for that order (assume that the questions are independent)?