Probability Theory
Q21.
Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick isQ22.
For any discrete random variable X, with probability mass function P(X=j)=p_{j}, p_{j}\geq 0, j \in \{0,....N\} and \sum_{j=0}^{N}p_{j}=1, define the polynomial function g_{x}(z)=\sum_{j=0}^{N}p_{j}z^{j}. For a certain discrete random variable Y, there exists a scalar \beta \in [0,1] such that g_{Y}(z)=(1-\beta +\beta z)^{N}. The expectation of Y isQ23.
Let S be a sample space and two mutually exclusive events A and B be such that A\cupB = S. If P(.) denotes the probability of the event, the maximum value of P(A)P(B) is ______Q24.
P and Q are considering to apply for a job. The probability that P applies for the job is 1/4. The probability that P applies for the job given that Q applies for the job is 1/2 , and the probability that Q applies for the job given that P applies for the job 1/3. Then the probability that P does not apply for the job given that Q does not apply for the job isQ25.
The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system is deemed functional be denoted by p. Then 100p= _____________.Q26.
Consider the following experiment. Step 1. Flip a fair coin twice. Step 2. If the outcomes are(TAILS, HEADS) then output Y and stop. Step 3. If the outcomes are either(HEADS, HEADS) or(HEADS, TAILS), then output N and stop. Step 4. If the out comes are(TAILS, TAILS), then go to Step1. The probability that the output of the experiment is Y is (up to two decimal places)_____.Q27.
The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is ______ .Q28.
For fair six-sided dice are rolled. The probability that the sum of the results being 22 is \frac{X}{1296}. The value of X is _________Q29.
Let P(E) denote the probability of the occurrence of event E. If P(A)= 0.5 and P(B)=1 then the values of P(A|B) and P(B|A) respectively areQ30.
Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?