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Recent questions and answers in Probability

25 votes
5 answers
1
We are given a set $X = \{X_1,\ldots,X_n\}$ where $X_i=2^i$. A sample $S\subseteq X$ is drawn by selecting each $X_i$ independently with probability $P_i = \frac{1}{2}$ . The expected value of the smallest number in sample $S$ is: $\left(\frac{1}{n}\right)$ $2$ $\sqrt n$ $n$
answered 1 day ago in Probability Soumya Saurav 4.3k views
60 votes
13 answers
2
Suppose a fair six-sided die is rolled once. If the value on the die is $1, 2,$ or $3,$ the die is rolled a second time. What is the probability that the sum total of values that turn up is at least $6$ ? $\dfrac{10}{21}$ $\dfrac{5}{12}$ $\dfrac{2}{3}$ $\dfrac{1}{6}$
answered Jul 11 in Probability ashishroushan4 13.2k views
22 votes
8 answers
3
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 ____
answered Jul 11 in Probability ashishroushan4 6.2k views
27 votes
4 answers
4
Seven (distinct) car accidents occurred in a week. What is the probability that they all occurred on the same day? $\dfrac{1}{7^7}\\$ $\dfrac{1}{7^6}\\$ $\dfrac{1}{2^7}\\$ $\dfrac{7}{2^7}\\$
answered Jul 11 in Probability ashishroushan4 8.7k views
11 votes
3 answers
5
0 votes
1 answer
6
A bag contains $80$ balls numbered {1, . . . , 80}. Before the game starts, you choose $10$ different numbers from among {1, . . . , 80} and write them on a piece of paper. Then $20$ balls are selected (without replacement) out of the bag at random. (a) What is the probability that all your numbers are selected?
answered Jun 6 in Probability sda 90 views
0 votes
1 answer
7
A bag contains $80$ balls numbered {1, . . . , 80}. Before the game starts, you choose $10$ different numbers from among {1, . . . , 80} and write them on a piece of paper. Then $20$ balls are selected (without replacement) out of the bag at random. (c) What is the probability that exactly $4$ of your numbers are selected?
answered Jun 6 in Probability sda 68 views
0 votes
1 answer
8
A bag contains $80$ balls numbered {1, . . . , 80}. Before the game starts, you choose $10$ different numbers from among {1, . . . , 80} and write them on a piece of paper. Then $20$ balls are selected (without replacement) out of the bag at random. (b) What is the probability that none of your numbers is selected?
answered Jun 6 in Probability sda 134 views
22 votes
7 answers
9
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 $\frac{^{2n}\mathrm{C}_n}{4^n}$ $\frac{^{2n}\mathrm{C}_n}{2^n}$ $\frac{1}{^{2n}\mathrm{C}_n}$ $\frac{1}{2}$
answered Jun 2 in Probability varunrajarathnam 4.7k views
50 votes
6 answers
10
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 $3$ $4$ $5$ $6$
answered Jun 2 in Probability varunrajarathnam 13.3k views
13 votes
3 answers
11
Suppose a box contains 20 balls: each ball has a distinct number in $\left\{1,\ldots,20\right\}$ written on it. We pick 10 balls (without replacement) uniformly at random and throw them out of the box. Then we check if the ball with number $``1"$ on it is present in the box. If it is ... that the ball with number $``2"$ on it is present in the box? $9/20$ $9/19$ $1/2$ $10/19$ None of the above
answered May 31 in Probability Yawar Rmir 1.4k views
12 votes
5 answers
12
$10\%$ of all email you receive is spam. Your spam filter is $90\%$ reliable: that is, $90\%$ of the mails it marks as spam are indeed spam and $90\%$ of spam mails are correctly labeled as spam. If you see a mail marked spam by your filter, what is the probability that it really is spam? $10\%$ $50\%$ $70\%$ $90\%$
answered May 7 in Probability soujanyareddy13 3.5k views
10 votes
3 answers
13
The $12$ houses on one side of a street are numbered with even numbers starting at $2$ and going up to $24$. A free newspaper is delivered on Monday to $3$ different houses chosen at random from these $12$. Find the probability that at least $2$ of these newspapers are delivered to houses with numbers strictly greater than $14$. $\frac{7}{11}$ $\frac{5}{12}$ $\frac{4}{11}$ $\frac{5}{22}$
answered May 7 in Probability soujanyareddy13 1.2k views
10 votes
3 answers
14
You arrive at a snack bar and you can't decide whether to order a lime juice or a lassi. You decide to throw a fair $6$-sided die to make the choice, as follows. If you throw $2$ or $6$ you order a lime juice. If you throw a $4$, you order a lassi. Otherwise, you throw the die ... is the probability that you end up ordering a lime juice? $\frac{1}{3}$ $\frac{1}{2}$ $\frac{2}{3}$ $\frac{3}{4}$
answered May 6 in Probability soujanyareddy13 670 views
2 votes
2 answers
15
Varsha lives alone and dislikes cooking, so she goes out for dinner every evening. She has two favourite restaurants, $\text{Dosa Paradise}$ and $\text{Kababs Unlimited}$, to which she travels by local train. The train to $\text{Dosa Paradise}$ runs every $10$ minutes, at $0, 10, 20, 30, 40$ ... ends up eating in $\text{Kababs Unlimited}$? $\frac{1}{5}$ $\frac{1}{3}$ $\frac{2}{5}$ $\frac{1}{2}$
answered May 6 in Probability soujanyareddy13 282 views
2 votes
2 answers
17
There are $7$ switches on a switchboard, some of which are on and some of which are off. In one move, you pick any $2$ switches and toggle each of them-if the switch you pick is currently off, you turn it on, if it is on, you turn it off. Your aim is to execute a sequence of moves and ... (off,on,off,on,off,off,on) (off,on,on,on,on,on,off) (on,off,on,on,on,on,on) (off,off,off,off,off,on,off)
answered May 4 in Probability soujanyareddy13 424 views
3 votes
2 answers
18
Let $C_{n}$ be the number of strings $w$ consisting of $n$ $X's$ and $n$ $Y's$ such that no initial segment of $w$ has more $Y's$ than $X's.$ Now consider the following problem. A person stands on the edge of a swimming pool holding a bag of $n$ red and $n$ blue balls. He draws a ball out ... $\frac{C_{n}}{\binom{2n}{n}}$ $\frac{n\cdot C_{n}}{(2n)!}$ $\frac{n\cdot C_{n}}{\binom{2n}{n}}$
answered May 4 in Probability soujanyareddy13 231 views
1 vote
2 answers
19
You are given two coins $A$ and $B$ that look identical. The probability that coin $A$ turns up heads is $\frac{1}{4}$, while the probability that coin $B$ turns up heads is $\frac{3}{4}.$ You choose one of the coins at random and toss it twice. If both the outcomes are heads, what is the probability that you chose coin $B?$ $\frac{1}{16}$ $\frac{1}{2}$ $\frac{9}{16}$ $\frac{9}{10}$
answered May 4 in Probability soujanyareddy13 308 views
2 votes
3 answers
20
Suppose you alternate between throwing a normal six-sided fair die and tossing a fair coin. You start by throwing the die. What is the probability that you will see a $5$ on the die before you see tails on the coin? $\frac{1}{12}$ $\frac{1}{6}$ $\frac{2}{9}$ $\frac{2}{7}$
answered May 4 in Probability soujanyareddy13 498 views
0 votes
2 answers
21
Oscar has lost his dog in either forest A (with a priori probability 0.4) or in forest B (with a priori probability 0.6). On any given day, if the dog is in A and Oscar spends a day searching for it in A, the conditional probability that he will find the dog that day is 0.25. Similarly, if the ... *0.25+(1/2)*0.15; but the answer given is=(0.5*0.4*0.25)+(0.5*0.6*0.15); what is wrong with my logic?
answered May 1 in Probability sda 479 views
3 votes
3 answers
22
A box contains 10 screws, 3 of which are defective.Two screws are drawn at random with replacement.The probability that none of the two screws is defective will be
answered Apr 29 in Probability shreyash_k 4.8k views
5 votes
3 answers
23
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 second question is not ... and then $\textsf{QuesA}$. Expected marks $22$. First $\textsf{QuesA}$ and then $\textsf{QuesB}$. Expected marks $16$.
answered Apr 3 in Probability Nikhil_dhama 1k views
4 votes
2 answers
24
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 probability of receiving $v$ ... $0.7$. If the received signal is $H,$ the probability that the transmitted signal was $H$ (rounded to $2$ decimal places) is __________.
answered Mar 31 in Probability Divyanshu Shukla 795 views
2 votes
2 answers
25
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 ____________.
answered Mar 31 in Probability Divyanshu Shukla 1.1k views
0 votes
2 answers
26
An airplane knows that 5 percent of the people making reservations on a certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up?
answered Mar 18 in Probability endurance1 1.6k views
1 vote
3 answers
27
You have a bag with $347$ black balls and $278$ white balls. Without looking, you pick up two balls from the bag and apply the following rule. If both balls are of the same colour, you throw them both away. Otherwise, you throw away the black ball and ... are possible, but the probability of it being white is greater. Both colours are possible, but the probability of it being black is greater.
answered Mar 17 in Probability Kabir5454 394 views
6 votes
4 answers
28
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 ...
answered Feb 21 in Probability Exynos 1.8k views
1 vote
1 answer
29
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 $X$ ... Both $S_1$ and $S_2$ 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
answered Feb 18 in Probability Keval Dodiya 791 views
4 votes
1 answer
30
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 _________
answered Feb 18 in Probability zxy123 836 views
30 votes
12 answers
31
Suppose we uniformly and randomly select a permutation from the $20 !$ permutations of $1, 2, 3\ldots ,20.$ What is the probability that $2$ appears at an earlier position than any other even number in the selected permutation? $\left(\dfrac{1}{2} \right)$ $\left(\dfrac{1}{10}\right)$ $\left(\dfrac{9!}{20!}\right)$ None of these
answered Jan 28 in Probability Nikhil_dhama 9k views
24 votes
4 answers
32
An examination paper has $150$ multiple choice questions of one mark each, with each question having four choices. Each incorrect answer fetches $-0.25$ marks. Suppose $1000$ students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is $0$ $2550$ $7525$ $9375$
answered Jan 27 in Probability Nikhil_dhama 5.5k views
25 votes
8 answers
33
Let $P(E)$ denote the probability of the event $E$. Given $P(A) = 1$, $P(B) =\dfrac{1}{2}$, the values of $P(A\mid B)$ and $P(B\mid A)$ respectively are $\left(\dfrac{1}{4}\right),\left(\dfrac{1}{2}\right)$ $\left(\dfrac{1}{2}\right),\left(\dfrac{1}{4}\right)$ $\left(\dfrac{1}{2}\right),{1}$ ${1},\left(\dfrac{1}{2}\right)$
answered Jan 27 in Probability Nikhil_dhama 6.4k views
3 votes
2 answers
34
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
answered Jan 23 in Probability ijnuhb 959 views
7 votes
6 answers
35
A spider is at the bottom of a cliff, and is $n$ inches from the top. Every step it takes brings it one inch closer to the top with probability $1/3$, and one inch away from the top with probability $2/3$, unless it is at the bottom in which case, it always gets one ... a function of $n$? It will never reach the top. Linear in $n$. Polynomial in $n$. Exponential in $n$. Double exponential in $n$.
answered Jan 22 in Probability Pascua 1.2k views
1 vote
2 answers
36
An airline operates a flight having 50 seats. As they expect some passenger to not show up, they overbook the flight by selling 51 tickets. The probability that an individual passenger will not show up is 0.01, independent of all other tourists. Each ticket costs Rs ... not available, the airline has to pay a compensation of Rs.1lakh to that passenger. What is the expected revenue of the airline?
answered Jan 20 in Probability susmit600 317 views
1 vote
1 answer
37
If $A$ and $B$ are two related events, and $P(A \mid B)$ represents the conditional probability, Bayes’ theorem states that $P(A\mid B) = \dfrac{P(A)}{P(B)} P(B\mid A)$ $P(A\mid B) = P(A) P(B) P(B\mid A)$ $P(A\mid B) = \dfrac{P(A)}{P(B)}$ $P(A\mid B) = P(A)+P(B)$
asked Apr 2, 2020 in Probability Lakshman Patel RJIT 284 views
0 votes
2 answers
38
1 vote
2 answers
39
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
asked Mar 31, 2020 in Probability Lakshman Patel RJIT 413 views
1 vote
2 answers
40
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 $P\left(Y \right)$ = ___________ . $0.54$ $0.66$ $0.33$ $0.44$
asked Mar 31, 2020 in Probability Lakshman Patel RJIT 774 views
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