Recent questions tagged probability

TIFR CSE 2021 | Part A | Question: 10
Lavanya and Ketak each flip a fair coin (i.e., both heads and tails have equal probability of appearing) $n$ times. What is the probability that Lavanya sees more heads than ketak? In the following, the binomial coefficient $\binom{n}{k}$ counts the number of $k$-element subsets of ... $\sum_{i=0}^{n}\frac{\binom{n}{i}}{2^{n}}$

768 views

soujanyareddy13 asked Mar 25, 2021

1 votes

1 answer

392

Lavanya and Ketak each flip a fair coin (i.e., both heads and tails have equal probability of appearing) $n$ times. What is the probability that Lavanya sees more heads t...

TIFR CSE 2021 | Part B | Question: 11
Suppose we toss a fair coin (i.e., both beads and tails have equal probability of appearing) repeatedly until the first time by which at least $\textit{two}$ heads and at least $\textit{two}$ tails have appeared in the sequence of tosses made. What is the expected number of coin tosses that we would have to make? $8$ $4$ $5.5$ $7.5$ $4.5$

555 views

soujanyareddy13 asked Mar 25, 2021

2 votes

2 answers

393

Suppose we toss a fair coin (i.e., both beads and tails have equal probability of appearing) repeatedly until the first time by which at least $\textit{two}$ heads and at...

GATE Civil 2021 Set 2 | GA Question: 3
Two identical cube shaped dice each with faces numbered $1$ to $6$ are rolled simultaneously. The probability that an even number is rolled out on each dice is: $\frac{1}{36}$ $\frac{1}{12}$ $\frac{1}{8}$ $\frac{1}{4}$

1.3k views

soujanyareddy13 asked Mar 25, 2021

4 votes

1 answer

394

Two identical cube shaped dice each with faces numbered $1$ to $6$ are rolled simultaneously. The probability that an even number is rolled out on each dice is:$\frac{1}{...

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2.4k views

go_editor asked Mar 1, 2021

3 votes

1 answer

395

GATE Mechanical 2021 Set 2 | GA Question: 7
A box contains $15$ blue balls and $45$ black balls. If $2$ balls are selected randomly, without replacement, the probability of an outcome in which the first selected is a blue ball and the second selected is a black ball, is _____ $\frac{3}{16}$ $\frac{45}{236}$ $\frac{1}{4}$ $\frac{3}{4}$

A box contains $15$ blue balls and $45$ black balls. If $2$ balls are selected randomly, without replacement, the probability of an outcome in which the first selected is...

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1.7k views

go_editor asked Mar 1, 2021

4 votes

1 answer

396

GATE Electrical 2021 | GA Question: 8
Let $X$ be a continuous random variable denoting the temperature measured. The range of temperature is $[0, 100]$ degree Celsius and let the probability density function of $X$ be $f\left ( x \right )=0.01$ for $0\leq X\leq 100$. The mean of $X$ is __________ $2.5$ $5.0$ $25.0$ $50.0$

Let $X$ be a continuous random variable denoting the temperature measured. The range of temperature is $[0, 100]$ degree Celsius and let the probability density function ...

GATE CSE 2021 Set 2 | Question: 22
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 _________

2.1k views

Arjun asked Feb 19, 2021

14 votes

1 answer

397

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 ...

GATE CSE 2021 Set 2 | Question: 29
In an examination, a student can choose the order in which two questions ($\textsf{QuesA}$ and $\textsf{QuesB}$) must be attempted. If the first question is answered wrong, the student gets zero marks. If the first question is answered correctly and the ... $22$. First $\textsf{QuesA}$ and then $\textsf{QuesB}$. Expected marks $16$.

6.1k views

Arjun asked Feb 18, 2021

27 votes

4 answers

398

In an examination, a student can choose the order in which two questions ($\textsf{QuesA}$ and $\textsf{QuesB}$) must be attempted.If the first question is answered wrong...

GATE CSE 2021 Set 2 | Question: 33
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 ...

7.5k views

Arjun asked Feb 18, 2021

36 votes

4 answers

399

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 ...

GATE CSE 2021 Set 1 | GA Question: 8
There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag. The probability that at least two chocolates are identical is __________ $0.3024$ $0.4235$ $0.6976$ $0.8125$

10.9k views

Arjun asked Feb 18, 2021

14 votes

1 answer

400

There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag.The probability that at least two chocolates are iden...

GATE CSE 2021 Set 1 | Question: 18
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 ____________.

10.8k views

Arjun asked Feb 18, 2021

16 votes

2 answers

401

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 pick...

GATE CSE 2021 Set 1 | Question: 35
Consider the two statements. $S_1:\quad$ There exist random variables $X$ and $Y$ such that $ \left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2>\textsf{Var}[X]\textsf{Var}[Y]$ $S_2:\quad$ For all random variables $X$ ... $S_2$ are true $S_1$ is true, but $S_2$ is false $S_1$ is false, but $S_2$ is true Both $S_1$ and $S_2$ are false

9.4k views

Arjun asked Feb 18, 2021

10 votes

1 answer

402

Consider the two statements.$S_1:\quad$ There exist random variables $X$ and $Y$ such that $ \left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2>\textsf{Var}[X]\t...

GATE CSE 2021 Set 1 | Question: 54
A sender $(\textsf{S})$ 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 $(\textsf{R})$. In the graph below, the weight of edge $(u,v)$ is the ... $0.7$. If the received signal is $H,$ the probability that the transmitted signal was $H$ (rounded to $2$ decimal places) is __________.

7.4k views

Arjun asked Feb 18, 2021

21 votes

2 answers

403

A sender $(\textsf{S})$ 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 $(\textsf{R})$.I...

GATE Overflow Test Series | Mock GATE | Test 5 | Question: 59
Suppose that you model the time between arrival of emails in your inbox as an exponentially distributed random variable with mean $21$ minutes. You get an email at $9:00$ am and then check your email at $9:21$ am. Unfortunately, ... what is the expected value of the number of minutes after $9:21$ am at which the next email will arrive?

5.6k views

Arjun asked Feb 18, 2021

10 votes

1 answer

404

Suppose that you model the time between arrival of emails in your inbox as an exponentially distributed random variable with mean $21$ minutes. You get an email at $9:00$...

GATE Overflow Test Series | Mock GATE | Test 5 | Question: 64
In the city of Rebuke birth rate is uniform for all months except September where the number of births is double the normal. Also, $3$ girl children are born in September for every $2$ boy children where as the ratio is the ... gender) is born in Rebuke what is the probability that her birthday is in September? (up to $3$ decimal places)

532 views

gatecse asked Feb 8, 2021

9 votes

1 answer

405

In the city of Rebuke birth rate is uniform for all months except September where the number of births is double the normal. Also, $3$ girl children are born in September...

GATE Overflow Test Series | Mock GATE | Test 4 | Question: 51
A fair die is thrown as long as necessary for $6$ to turn up. If the number of throws $(n)$ turns out to be odd, the probability that $n=5$ is ______ (rounded to three decimal places)

631 views

gatecse asked Feb 8, 2021

3 votes

1 answer

406

A fair die is thrown as long as necessary for $6$ to turn up. If the number of throws $(n)$ turns out to be odd, the probability that $n=5$ is ______ (rounded to three de...

GATE Overflow Test Series | Mock GATE | Test 4 | Question: 54
A certain family has $6$ children, consisting of $3$ boys and $3$ girls. Assuming that all birth orders are equally likely with no twins, what is the probability that the $2$ eldest children are girls? (Up to $2$ decimal places)

509 views

gatecse asked Feb 1, 2021

1 votes

1 answer

407

A certain family has $6$ children, consisting of $3$ boys and $3$ girls. Assuming that all birth orders are equally likely with no twins, what is the probability that the...

CMI-2018-DataScience-A: 5
$\text{Description for the following question:}$ If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then $\mathbb{P}(Z\leq a)=\int^a _{-\infty} \frac{\exp\{\frac{-z^2}{2}\}}{\sqrt{2\pi}}dz=\Phi(a), $ ... .02 $\int^{18} _{-\infty} \frac{1}{\sqrt{2\pi .2^2}}\exp\{-\frac{1}{2}(\frac{x-24}{2})^2\}dx$

407 views

gatecse asked Feb 1, 2021

0 votes

1 answer

408

$\text{Description for the following question:}$If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then$$\...

CMI-2018-DataScience-A: 6
$\text{Description for the following question:}$ If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then $\mathbb{P}(Z\leq a)=\int^a _{-\infty} \frac{\exp\{\frac{-z^2}{2}\}}{\sqrt{2\pi}}dz=\Phi(a), $ ... more than 0.4 less than 0.5

595 views

soujanyareddy13 asked Jan 29, 2021

2 votes

0 answers

409

$\text{Description for the following question:}$If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then$$\...

CMI-2018-DataScience-A: 7
$\text{Description for the following question:}$ If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$ ... than $26$ months approximately equals $16\%$ is more than $15\%$ is less than $14\%$ is between $10\%$ and $15\%$

382 views

soujanyareddy13 asked Jan 29, 2021

0 votes

0 answers

410

$\text{Description for the following question:}$If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then$$...

CMI-2018-DataScience-A: 12
In an entrance examination with multiple choice questions, with each question having four options and a single correct answer, suppose that only $20\%$ candidates think they know the answer to one difficult question and only half of them know it ... same. If a candidate has correctly answered the question, what is the (conditional) probability that she knew the answer?

284 views

soujanyareddy13 asked Jan 29, 2021

0 votes

2 answers

411

In an entrance examination with multiple choice questions, with each question having four options and a single correct answer, suppose that only $20\%$ candidates think t...

CMI-2018-DataScience-B: 8
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the ... credit default in a year. You can assume that whether a given debtor will default or not is independent of the behavior of other debtors.

923 views

soujanyareddy13 asked Jan 29, 2021

0 votes

1 answer

412

For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{...

CMI-2018-DataScience-B: 9
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number ... expected number of success is $\mathbb{E}(X)=np$. For the situation in the previous problem, what is the expected number defaults?

247 views

soujanyareddy13 asked Jan 29, 2021

0 votes

1 answer

413

For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc.$\text{D...

CMI-2018-DataScience-B: 10
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number of ... credit than the bank loses the entire loan amount. What is the expected revenue of the bank from a loan of $Rs. 100,000?$

220 views

soujanyareddy13 asked Jan 29, 2021

0 votes

1 answer

414

For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc.$\text{D...

CMI-2020-DataScience-A: 14
Suppose you roll two six-sided fair dice with faces numbered from $1$ to $6$ and take the sum of the two numbers that turn up. What is the probability that: the sum is $12;$ the sum is $12$, given that the sum is even; the sum is $12$, given that the sum is an ... $\frac {1}{14}$, respectively $\frac {1}{36}, \frac {1}{16} $, and $\frac {1}{12}$, respectively

208 views

soujanyareddy13 asked Jan 29, 2021

0 votes

1 answer

415

Suppose you roll two six-sided fair dice with faces numbered from $1$ to $6$ and take the sum of the two numbers that turn up. What is the probability that:the sum is $12...

GATE Overflow Test Series | Mock GATE | Test 3 | Question: 45
A delivery company divides their packages into weight classes. Suppose packages in the $13$ to $21$ kilogram class are uniformly distributed, meaning that all weights within that class are equally likely to occur. If the probability that a ... Then the value of $\alpha - \beta + \gamma - \rho$ is _________ (upto two decimal places)

188 views

soujanyareddy13 asked Jan 29, 2021

4 votes

1 answer

416

A delivery company divides their packages into weight classes. Suppose packages in the $13$ to $21$ kilogram class are uniformly distributed, meaning that all weights wit...

GATE Overflow Test Series | Mock GATE | Test 3 | Question: 61
Two coins are in a box. The coins look alike, but one coin is fair (with equal probabilities for Head and Tail), while the other coin is biased, with $2/3$ probability of Heads. One of the coins is randomly chosen from the hat, without knowing which ... chosen coin is the fair coin? $\dfrac{4}{5}$ $0.692$ $\dfrac{9}{25}$ $\dfrac{1}{2}$

523 views

gatecse asked Jan 26, 2021

5 votes

1 answer

417

Two coins are in a box. The coins look alike, but one coin is fair (with equal probabilities for Head and Tail), while the other coin is biased, with $2/3$ probability of...

GATE Overflow Test Series | Mock GATE | Test 2 | Question: 30
A player tosses two fair coins. He wins $\$2$ if two heads occur, and $\$1$ if one head occurs. On the other hand, he loses $\$3$ if no heads occur. Find the expected value $E$ of the game. $1$ $0.5$ $0.25$ $0.75$

272 views

gatecse asked Jan 26, 2021

3 votes

1 answer

418

A player tosses two fair coins. He wins $\$2$ if two heads occur, and $\$1$ if one head occurs. On the other hand, he loses $\$3$ if no heads occur. Find the expected val...

GATE Overflow Test Series | Mock GATE | Test 1 | Question: 58
A school has three divisions-- $A,B$ and $C$ for the tenth grade, having the number of students $40, 45$ and $42$ respectively. The pass percentages of students from the three divisions are $96, 97$ and $98$ respectively. If a ... in the exam, what is the probability that it he/she is from the $A$ grade? (Rounded to $2$ decimal points)

306 views

gatecse asked Jan 17, 2021

5 votes

1 answer

419

A school has three divisions $A,B$ and $C$ for the tenth grade, having the number of students $40, 45$ and $42$ respectively. The pass percentages of students from the t...

NIELIT Scientific Assistant A 2020 November: 99
A bag contains $10$ white balls and $5$ blue balls. A ball is drawn from the bag and its color is noted. This ball is put back in the bag along with $3$ more balls of the same color. A ball is drawn again from the bag at random. The probability that the first ball drawn is blue, given that the second ball drawn is blue, is: $1/3$ $3/4$ $8/9$ $4/9$

578 views

gatecse asked Jan 3, 2021

2 votes

1 answer

420

A bag contains $10$ white balls and $5$ blue balls. A ball is drawn from the bag and its color is noted. This ball is put back in the bag along with $3$ more balls of the...