Probability Theory
Q11.
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) _________.Q12.
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:Q13.
Two people, P and Q, decide to independently roll two identical dice, each with 6 faces, numbered 1 to 6. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by P and Q. Assume that all 6 numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to 3 decimal places) that one of them wins on the third trial isQ14.
Suppose X_{i} for i =1,2,3 are independent and identically distributed random variables whose probability mass functions are Pr[X_{i}=0]=Pr[X_{i}=1]=1/2 for i=1,2,3. Define another random variable Y=X_{1}X_{2}\oplus X_{3}, \; where \; \oplus denotes XOR. Then Pr[Y=0|X_{3}=0]=_______________.Q15.
Let A and B be any two arbitrary events, then, which one of the following is TRUE?Q16.
Each of the nine words in the sentence "The quick brown fox jumps over the lazy og" is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____________. (The answer should be rounded to one decimal place)Q17.
A probability density function on the interval [a,1] is given by 1/x^{2} and outside this interval the value of the function is zero.The value of a is __________.Q19.
Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is _________.Q20.
Let X be a Gaussian random variable mean 0 and variance \sigma ^{2} . Let Y=max(X, 0) where max (a,b) is the maximum of a and b. The median of Y is ____________.