# Recent questions and answers in Probability

1
Consider a company that assembles computers. The probability of a faulty assembly of any computer is $p$. The company therefore subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of $q$. What is the probability of a computer being declared faulty? $pq + (1 - p)(1 - q)$ $(1 - q)p$ $(1 - p)q$ $pq$
2
A deck of $5$ cards (each carrying a distinct number from $1$ to $5$) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the two cards are selected with the number on the first card being one higher than the number on the second card? ... $\left(\dfrac{4}{25}\right)$ $\left(\dfrac{1}{4}\right)$ $\left(\dfrac{2}{5}\right)$
3
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 _________.
4
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 _________.
5
$P$ and $Q$ are considering to apply for a job. The probability that $P$ applies for the job is $\dfrac{1}{4},$ the probability that $P$ applies for the job given that $Q$ applies for the job is $\dfrac{1}{2},$ and the probability that $Q$ applies for the job given that $P$ applies ... $\left(\dfrac{4}{5}\right)$ $\left(\dfrac{5}{6}\right)$ $\left(\dfrac{7}{8}\right)$ $\left(\dfrac{11}{12}\right)$
6
For any discrete random variable $X$, with probability mass function $P(X=j)=p_j, p_j \geq 0, j \in \{0, \dots , N \}$, and $\Sigma_{j=0}^N \: p_j =1$, define the polynomial function $g_x(z) = \Sigma_{j=0}^N \: p_j \: z^j$. For a certain discrete random variable $Y$, ... $Y$ is $N \beta(1-\beta)$ $N \beta$ $N (1-\beta)$ Not expressible in terms of $N$ and $\beta$ alone
7
The probability of an event $B$ is $P_1$. The probability that events $A$ and $B$ occur together is $P_2$ while the probability that $A$ and $\bar{B}$ occur together is $P_3$. The probability of the event $A$ in terms of $P_1, P_2$ and $P_3$ is _____________
1 vote
8
A arrives at office at 8-10am regularly; B arrives at 9-11 am every day. Probability that one day B arrives before A? [Assume arrival time of both A and B are uniformly distributed]
9
An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the trials are independent, the expected number of tosses is $3$ $4$ $5$ $6$
10
Two $n$ bit binary strings, $S_1$ and $S_2$ are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to $d$ is $\dfrac{^{n}C_{d}}{2^{n}}$ $\dfrac{^{n}C_{d}}{2^{d}}$ $\dfrac{d}{2^{n}}$ $\dfrac{1}{2^{d}}$
11
A random bit string of length n is constructed by tossing a fair coin n times and setting a bit to 0 or 1 depending on outcomes head and tail, respectively. The probability that two such randomly generated strings are not identical is: $\frac{1}{2^n}$ $1 - \frac{1}{n}$ $\frac{1}{n!}$ $1 - \frac{1}{2^n}$
1 vote
12
A lottery chooses four random winners. What is the probability that at least three of them are born on the same day of the week? Assume that the pool of candidates is so large that each winner is equally likely to be born on any of the seven days of the week independent of the other winners. ... $\dfrac{48}{2401} \\$ $\dfrac{105}{2401} \\$ $\dfrac{175}{2401} \\$ $\dfrac{294}{2401}$
13
A fair coin is tossed n times .find probability of difference between head and tails is n-3
1 vote
14
Two balls are drawn uniformly at random without replacement from a set of five balls numbered $1,2,3,4,5.$ What is the expected value of the larger number on the balls drawn? $2.5$ $3$ $3.5$ $4$ None of the above
15
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 ... equi-probable and that all trials are independent. The probability (rounded to $3$ decimal places) that one of them wins on the third trial is ____
16
Consider a random variable $X$ that takes values $+1$ and $−1$ with probability $0.5$ each. The values of the cumulative distribution function $F(x)$ at $x = −1$ and $+1$ are $0$ and $0.5$ $0$ and $1$ $0.5$ and $1$ $0.25$ and $0.75$
17
A point is randomly selected with uniform probability in the $X-Y$ plane within the rectangle with corners at $(0,0), (1,0), (1,2)$ and $(0,2).$ If $p$ is the length of the position vector of the point, the expected value of $p^{2}$ is $\left(\dfrac{2}{3}\right)$ $\quad 1$ $\left(\dfrac{4}{3}\right)$ $\left(\dfrac{5}{3}\right)$
18
A box contains $10$ screws, $3$ of which are defective. Two screws are drawn at random with replacement. The probability that none of two screws is defective will be $100\%$ $50\%$ $49\%$ None of these.
19
For each element in a set of size $2n$, an unbiased coin is tossed. The $2n$ coin tosses are independent. An element is chosen if the corresponding coin toss was a head. The probability that exactly $n$ elements are chosen is $\frac{^{2n}\mathrm{C}_n}{4^n}$ $\frac{^{2n}\mathrm{C}_n}{2^n}$ $\frac{1}{^{2n}\mathrm{C}_n}$ $\frac{1}{2}$
20
A box contains $5$ fair and $5$ biased coins. Each biased coin has a probability of head $\frac{4}{5}$. A coin is drawn at random from the box and tossed. Then the second coin is drawn at random from the box ( without replacing the first one). Given that the first coin has shown head, the ... that the second coin is fair is $\frac{20}{39}\\$ $\frac{20}{37}\\$ $\frac{1}{2}\\$ $\frac{7}{13}$
1 vote
21
In a sample of 100 students, the mean of the marks (only integers) obtained by them in a test is 14 with its standard deviation of 2.5(marks obtained can be fitted with a normal distribution ).the percentage of students scoring 16 marks is a)36 b)23 c)12 d)10 (Area under standard normal curve between z=0 and z=0.6 is 0.2257 ; and between z=0 and z=1.0 is 0.3413)
1 vote
22
In a bunch of $13$ T-shirts only $1$ is of Medium size, which is correct fit for the searching person. Each time wrong size is picked, the person throws it away and pick the next T-shirt. What is the probability that the correct size T-shirt can be searched in $8^{th}$ attempt ? My attempt : $\frac{1}{13}$ where i went wrong ?
23
A bag contains $10$ white balls and $15$ black balls. Two balls are drawn in succession. The probability that one of them is black and the other is white is: $\frac{2}{3}$ $\frac{4}{5}$ $\frac{1}{2}$ $\frac{1}{3}$
1 vote
24
Let $X_1$, and $X_2$ and $X_3$ be chosen independently from the set $\{0, 1, 2, 3, 4\}$, each value being equally likely. What is the probability that the arithmetic mean of $X_1, X_2$ and $X_3$ is the same as their geometric mean? $\frac{1}{5^2}\\$ $\frac{1}{5^3}\\$ $\frac{3!}{5^3}\\$ $\frac{3}{5^3}$
25
The probability that a given positive integer lying between $1$ and $100$ (both inclusive) is NOT divisible by $2$, $3$ or $5$ is ______ .
26
Suppose $X$ and $Y$ are two independent random variables both following Poisson distribution with parameter $\lambda$. What is the value of $E(X-Y)^2$ ? $\lambda$ $2 \lambda$ $\lambda^2$ $4 \lambda^2$
27
What is the probability that a Normal random variable differs from its mean $\mu$ by more than $\sigma$ ?
28
A drawer contains $2$ Blue, $4$ Red and $2$ Yellow balls. No two balls have the same radius. If two balls are randomly selected from the drawer, what is the probability that they will be of the same colour? $\left(\dfrac{2}{7}\right)$ $\left(\dfrac{2}{5}\right)$ $\left(\dfrac{3}{7}\right)$ $\left(\dfrac{1}{2}\right)$ $\left(\dfrac{3}{5}\right)$
29
You have to play three games with opponents $A$ and $B$ in a specified sequence. You win the series if you win two consecutive games. $A$ is a stronger player than $B$. Which sequence maximizes your chance of winning the series? $AAB$ $ABA$ $BAB$ $BAA$ All are the same.
1 vote
30
If $X, Y$ and $Z$ are three exhaustive and mutually exclusive events related with any experiment and the $P\left(X \right)=0.5P\left(Y \right)$ and $P\left(Z \right)$ = $0.3P\left(Y \right)$. Then $P\left(Y \right)$ = ___________ . $0.54$ $0.66$ $0.33$ $0.44$
1 vote
31
If a random coin is tossed $11$ times, then what is the probability that for $7$th toss head appears exactly $4$ times? $5/32$ $15/128$ $35/128$ None of the options
1 vote
32
Let x be the number obtained from rolling a fair dice and you toss an unbiassed coin X times. What is the probablity that X=5 given that you have obtained 3 heads from X tosses?
33
In a certain year, there were exactly four Fridays and exactly four Mondays in January. On what day of the week did the $20^{th}$ of the January fall that year (recall that January has $31$ days)? Sunday Monday Wednesday Friday None of the others
34
If $P$ is risk probability, $L$ is loss, then Risk Exposure $(RE)$ is computed as. $RE = P/L$ $RE = P + L$ $RE = P \ast L$ $RE = 2 \ast P \ast L$
35
Candidates were asked to come to an interview with 3 pens each. Black, blue, green and red were the permitted pen colours that the candidate could bring. The probability that a candidate comes with all 3 pens having the same colour is _________.
1 vote
36
The probability that top and bottom cards of a randomly shuffled deck are both aces is: $4/52\times 4/52$ $4/52\times 3/52$ $4/52\times 3/51$ $4/52\times 4/51$
37
Ram has a fair coin, i.e., a toss of the coin results in either head or tail and each event happens with probability exactly half $(1/2)$. He repeatedly tosses the coin until he gets heads in two consecutive tosses. The expected number of coin tosses that Ram does is. $2$ $4$ $6$ $8$ None of the above.
1 vote
A box contains six red balls and four green balls. Four balls are selected at random from the box. What is the probability that two of the selected balls are red and two are green ? $\large\frac{3}{7}$ $\large\frac{4}{7}$ $\large\frac{5}{7}$ $\large\frac{6}{7}$
If $A_1, A_2, \dots , A_n$ are independent events with probabilities $p_1, p_2, \dots , p_n$ respectively, then $P( \cup_{i=1}^n A_i)$ equals $\Sigma_{i=1}^n \: \: p_i$ $\Pi_{i=1}^n \: \: p_i$ $\Pi_{i=1}^n \: \: (1-p_i)$ $1-\Pi_{i=1}^n \: \: (1-p_i)$