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Recent questions tagged probability

1 1 vote
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GO Classes DPP | GATE DA | Probability | Probability Rules & Properties of Events
Which of the following statements is NOT always true?If $A \subseteq B$, then $P(B \mid A)=P(B)-P(A)$. If $P(B)>0$ and $P(A \mid B)=P(A)$, then $A$ and $B$ are independen...
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5 5 votes
2 2 answers
208
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GO Classes DPP | GATE DA | Probability | Conditional Probability
Let $X \in {0,1}$ and $Y \in {0,1}$ be two independent binary random variables. If $P(X=1)=p$ and $P(Y=1)=q$, then $P(X+Y=1 \mid X+Y \ge 1)$ is equal to$\dfrac{p+q-2pq}{p...
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4 4 votes
2 2 answers
203
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GO Classes DPP | GATE DA | Probability | Discrete Random Variables
Let $X$ be a random variable that takes values $-1$, $0$, and $2$, with $P(X=-1)=p$, $P(X=0)=2p$, and $P(X=2)=1-3p$. If $E[X]=\frac{1}{2}$, what is $E[X^2]$?$\frac{19}{14...
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4 4 votes
2 2 answers
201
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GO Classes DPP | GATE DA | Probability | Probability with Replacement
A bag contains $6$ blue marbles, $9$ green marbles and $15$ red marbles. A marble is drawn from the bag, its colour is recorded and it is put back in the bag. This proces...
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5 5 votes
3 3 answers
261
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GO Classes DPP | GATE DA | Probability | De Morgan’s Laws
A sample space has two events $A$ and $B$ such that $P(A' \cap B')=\frac{1}{8}$, $P(A' \cup B)=\frac{5}{6}$, and $P(A \cup B')=\frac{3}{4}$. What is $P(A \cap B)$?$\frac{...
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3 3 votes
3 3 answers
282
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GO Classes DPP | GATE DA | Probability | Probability Density Function
Let $x$ be a random variable possessing the probability density function$$f(x)= \begin{cases}c x & , x \in[0,10] \\\\ 0 & , \text { otherwise }\end{cases}$$where $c \in \...
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4 4 votes
3 3 answers
253
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GO Classes DPP | GATE DA | Probability | Geometric Distribution
Deepak Poonia Sir is playing a game that involves repeatedly rolling a fair standard $6$-sided die. What is the expected number of rolls until he gets a $5$ ?$5$ $6$ $1/6...
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3 3 votes
2 2 answers
223
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GO Classes DPP | GATE DA | Probability | Random Variable
Let $X$ be a random variable that takes values from $0$ to $9$ with equal probability $\frac{1}{10}$. Define the random variable $Y=X \bmod 3$. Calculate $P(Y=0)+P(Y=1)$....
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2 2 votes
2 2 answers
208
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GO Classes DPP | GATE DA | Probability | Discrete Random Variables
Suppose $X$ is the number of dust storms that occur on Mars next year. Assume that $X$ is a discrete uniform random variable that takes one of the $101$ values in the ran...
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2 2 votes
2 2 answers
223
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GO Classes DPP | GATE DA | Probability | Probability Distribution
The following is the probability distribution for a random variable, $X$.$$\begin{array}{c|c}k & P(X = k) \\\hline0 & 0.3 \\1 & 0.5 \\2 & 0.1 \\3 & 0.1\end{array}$$What i...
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0 0 votes
2 2 answers
265
265 views
GO Classes DPP | GATE DA | Probability | Inclusion-Exclusion Principle
In a sample space, two events $A$ and $B$ satisfy $P(A') = \frac{1}{4}$, $P(B') = \frac{1}{3}$, and $P(A \cap B) = \frac{1}{2}$. What is $P(A \cup B)$?$\dfrac{7}{12}$ $\d...
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1 1 vote
3 3 answers
217
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GO Classes DPP | GATE DA | Probability | Counting Methods
Assume that birthdays are uniformly distributed over $365$ days of the year and ignore leap years. In a randomly chosen group of $n$ people, what is the probability that ...
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2 2 votes
2 2 answers
239
239 views
GO Classes DPP | GATE DA | Probability | Law of Total Probability
A bag contains two biased coins. Coin A shows heads with probability $\frac{3}{4}$, and Coin B shows heads with probability $\frac{1}{3}$. One of the two coins is chosen ...
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3 3 votes
2 2 answers
212
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GO Classes DPP | GATE DA | Probability | Conditional Probability
In a certain city, the probability that a person will carry an umbrella on a given day is $0.5$. It is known that if the morning is cloudy, then the probability that the ...
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1 1 vote
2 2 answers
204
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GO Classes DPP | GATE DA | Probability | Discrete Random Variable
A biased die is rolled repeatedly. On each roll, the probability of getting a $6$ is $p$, where $0 < p < 1$. Let $X$ be the random variable denoting the number of rolls u...
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3 3 votes
2 2 answers
254
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GO Classes DPP | GATE DA | Probability | Mutually Exclusive & Exhaustive Events
Let $S$ be a sample space and let $A$, $B$, and $C$ be three mutually exclusive events such that $A \cup B \cup C = S.$If $P(A)=x$, $P(B)=2x$, and $P(C)=1-3x$, then the m...
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3 3 votes
4 4 answers
289
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GO Classes DPP | GATE DA | Probability | Conditional Probability
Three fair coins are flipped. Given that at least one of the coins shows a head and at least one of the coins shows a tail, what is the probability that exactly two of th...
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3 3 votes
3 3 answers
233
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GO Classes DPP | GATE DA | Probability | Selection Without Replacement
A deck of $6$ cards, each carrying a distinct number from $1$ to $6$, is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probabil...
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5 5 votes
2 2 answers
207
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GO Classes DPP | GATE DA | Probability | Conditional Probability
A medical laboratory buys testing kits from two manufacturers, $A$ and $B$. Manufacturer $A$ supplies $65\%$ of the kits, and manufacturer $B$ supplies $35\%$ of the kits...
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4 4 votes
2 2 answers
262
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GO Classes DPP | GATE DA | Probability | Inclusion-Exclusion Principle
The probability that a student passes Algebra is $0.7$. The probability that the student passes Biology is $0.6$. The probability that the student passes Chemistry is $0....
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3 3 votes
2 2 answers
227
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GO Classes DPP | GATE DA | Probability | Combinatorial Probability
Sixteen players $P_1, P_2, P_3, \ldots, P_{16}$ participate in a knockout tournament. In any match between $P_i$ and $P_j$, player $P_i$ wins if $i<j$. Assume that the po...
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4 4 votes
3 3 answers
218
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GO Classes DPP | GATE DA | Probability | Binomial Distribution
Suppose $n$ balls are randomly distributed into $K$ compartments. Each compartment can contain any number of balls from $0$ to $n$. Assuming that all $K^n$ arrangements a...
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3 3 votes
2 2 answers
190
190 views
GO Classes DPP | GATE DA | Probability | Selection Without Replacement
A forest contains $20$ elk, of which $5$ are captured, tagged, and released. A week later, $4$ of the $20$ elk are captured again. What is the probability that exactly $2...
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3 3 votes
2 2 answers
173
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GO Classes DPP | GATE DA | Probability | Combinatorial Counting or Enumeration
Two fair dice are rolled, one at a time. What is the probability that the second die will show a number higher than showed on the first die?$\frac{5}{12}$ $\frac{7}{12}$ ...
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3 3 votes
2 2 answers
229
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GO Classes DPP | GATE DA | Probability | Conditional Probability
One hundred people line up to board an airplane. Each has a boarding pass with assigned seat. However, the first person to board has lost his boarding pass and takes a ra...
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5 5 votes
4 4 answers
371
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GO Classes DPP | GATE DA | Probability | Conditional Probability
There are three doors; behind one is a nice car, and behind each of the other two is a goat eating a bale of straw. You choose a door. Then Monty Hall opens one of the ot...
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7 7 votes
3 3 answers
355
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GO Classes DPP | GATE DA | Probability | Conditional Probability
A multiple choice exam has $4$ choices for each question. The student has studied enough so that the probability they will know the answer to a question is $0.5$, the pro...
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3 3 votes
2 2 answers
250
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GO Classes DPP | GATE DA | Probability | Conditional Probability
Sarah and Bob draw $13$ cards each from a standard deck of $52$ cards. Given that Sarah has exactly two aces, what is the probability that Bob has exactly one ace?$\dfrac...
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3 3 votes
3 3 answers
257
257 views
GO Classes DPP | GATE DA | Probability | Conditional Probability
Consider a clinical test for cancer that can yield either a positive $(+)$ or negative $(-)$ result. Suppose that a patient who truly has cancer has a $1\%$ chance of sli...
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3 3 votes
2 2 answers
262
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GO Classes DPP | GATE DA | Probability | Conditional Probability
Mark all expressions which indicate that $\mathrm{X}$ is independent of $\mathrm{Y}$ given $\mathrm{Z}$.$P(X, Y \mid Z)=P(X \mid Z) P(Y \mid Z) $ $P(X, Y, Z)=P(X, Z) P(Y)...
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