Probability Theory

A bag has r red balls and b black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the ball is placed back into the bag along with another ball of the same colour. Note that the number of balls in the bag will increase by one, after the trial. A sequence of four such trials is conducted. Which one of the following choices gives the probability of drawing a red ball in the fourth trial?

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.

\begin{array}{c|c|c} \text{question} & \text{probabiloty of answering correctly} & \text{marks} \\ \hline \textsf{QuesA} & 0.8 & 10 \\ \textsf{QuesB} & 0.5 & 20 \end{array}

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)?

For a given biased coin, the probability that the outcome of a toss is a head is 0.4. This coin is tossed 1,000 times. Let X denote the random variable whose value is the number of times that head appeared in these 1,000 tosses. The standard deviation of X (rounded to 2 decimal place) is ________

A sender (S) transmits a signal, which can be one of the two kinds: H and L with probabilities 0.1 and 0.9 respectively, to a receiver (R)
In the graph below, the weight of edge (u,v) is the probability of receiving v when u is transmitted, where u,v\in\{H,L\}. For example, the probability that the received signal is L given the transmitted signal was H, is 0.7.

If the received signal is H, the probability that the transmitted signal was H (rounded to 2 decimal places) is __________.

The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 2. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to 2 decimal places) is _________

For the distributions given below :

Which of the following is correct for the above distributions?

For n\gt2, let a \in \{0,1\}^n be a non-zero vector. Suppose that x is chosen uniformly at random from \{0,1\}^n. Then, the probability that \sum_{i=1}^{n}a_ix_i is an odd number is______

Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x^2+6xY+3Y+6 has only real roots is (rounded off to 1 decimal place) _________.

Two numbers are chosen independently and uniformly at random from the set {1, 2, ..., 13}. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary representations have the same most significant bit is ___________

A class of 30 students occupy a classroom containing 5 rows of seats, with 8 seats in each row. If the students seat themselves at random, the probability that sixth seat in the fifth row will be empty is: