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Recent questions tagged probability

4 4 votes
1 1 answer
193
193 views
GO Classes DPP | GATE CS, DA | Probability | Conditional Probability
Suppose that $B_1, B_2, \ldots, B_k$ form a partition of the sample space $S$, and $A$ is an event with $0 < P(A) < 1$. Assume also that $P(B_i) 0$ for all $i$. Which of...
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3 3 votes
2 2 answers
210
210 views
GO Classes DPP | GATE CS, DA | Probability | Conditional Probability
Which of the following statements is/are always correct?If $A$ and $B$ are independent and $P(B) 0$, then $P(A^c \mid B) = 1 - P(A)$ If $A$ and $B$ are mutually exclusiv...
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2 2 votes
1 1 answer
171
171 views
GO Classes DPP | GATE CS, DA | Probability | Events in Probability
Let $S$ be the sample space. Three events $A$, $B$, and $C$ are such that $A$ and $C$ do not overlap, $C \subseteq B$, and $A$ overlaps with $B$. Which of the following s...
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4 4 votes
1 1 answer
183
183 views
GO Classes DPP | GATE CS | Probability | Combinatorial Probability
A fair six-sided die is rolled $4$ times. What is the probability that exactly $2$ of the outcomes are odd and the sum of the $4$ outcomes is $14$?$\frac{5}{54}$ $\frac{1...
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1 1 vote
1 1 answer
205
205 views
GO Classes DPP | GATE DA | Probability | Random Variables
A number between $0$ and $1$, inclusive, is chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is $...
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2 2 votes
2 2 answers
162
162 views
GO Classes DPP | GATE CS | Probability | Counting & Inclusion-Exclusion Principle
A game machine releases $6$ tokens, and each token shows a random integer from $1$ to $6$ with equal probability. The probability that the total of the $6$ numbers shown ...
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1 1 vote
1 1 answer
157
157 views
GO Classes DPP | GATE CS | Probability | Probability based on Combinations
A debate club has $14$ members, of which exactly $6$ have completed advanced training. A coach interviews $4$ members selected at random without replacement, one by one. ...
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2 2 votes
1 1 answer
170
170 views
GO Classes DPP | GATE DA | Probability | Variance of Discrete Random Variable
Suppose that $X$ is a random variable where $P(X=-4)=c$, $P(X=-1)=2c$, $P(X=2)=3c$, and $P(X=5)=4c$, where $c>0$. Let $Y=(X-E[X])^2$. What is the value of $E[Y]$?$7$ $8$ ...
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2 2 votes
1 1 answer
145
145 views
GO Classes DPP | GATE DA | Probability | Continuous Random Variables
Let $X$ be a continuous random variable with probability density function$$f(x)= \begin{cases}cx^2& , x \in[0,1] \\\\ c(2-x) & , x \in(1,2]\\\\0 & , \text { otherwise }\e...
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3 3 votes
3 3 answers
198
198 views
GO Classes DPP | GATE CS | Probability | Discrete Random Variables
Let $X$ be a random variable that takes each integer value from $1$ to $30$ with equal probability $\frac{1}{30}$. Define the random variable $Y = X \bmod 7$. Calculate $...
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2 2 votes
2 2 answers
158
158 views
GO Classes DPP | GATE DA | Probability | Markov Chains
James Bond repeatedly rolls a fair standard $6$-sided die. What is the expected number of rolls until he gets two consecutive $5$'s for the first time?$36$ $40$ $42$ $48$
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2 2 votes
1 1 answer
128
128 views
GO Classes DPP | GATE DA | Probability | Discrete Random Variables
In the experiment of tossing two fair dice, let $X$ be the larger of the two top faces. Which of the following gives the probability distribution of $X$?$x_k = k,\ p_k = ...
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1 1 vote
2 2 answers
175
175 views
GO Classes DPP | GATE CS | Probability | Inclusion-Exclusion Principle
Among $60$ students in a class, $28$ earned an $A$ on the midterm, $25$ earned an $A$ on the final exam, and $21$ earned an $A$ on the project. Also, $11$ earned an $A$ o...
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2 2 votes
1 1 answer
132
132 views
GO Classes DPP | GATE DA | Probability | Bernoulli Trials
Consider a coin with probability $p$ of showing heads on each toss. The coin is tossed repeatedly until the second head appears. Find the probability that the first head ...
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2 2 votes
1 1 answer
118
118 views
GO Classes DPP | GATE DA | Probability | Continuous Random Variables
Let $X$ be a continuous random variable with the probability density function $$f(x)= \begin{cases} c\left(2-|x|\right), & -1\lt x\lt 2 \\\\ 0, & \text { otherwise }\end{...
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1 1 vote
1 1 answer
111
111 views
GO Classes DPP | GATE DA | Probability | Discrete Probability
Five candidates, $A$, $B$, $C$, $D$, and $E$, are running for the same office, and exactly one of them wins. The sample space is $\Omega={A,B,C,D,E}$. Suppose $A$ and $B$...
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1 1 vote
2 2 answers
162
162 views
GO Classes DPP | GATE DA | Probability | Probability Identities
Suppose $A$ and $B$ are independent events with $0<P(A)<1$ and $0<P(B)<1$. Which of the following statements is/are always correct?$P(A^c\cap B^c)=(1-P(A))(1-P(B))$ $P(A\...
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1 1 vote
1 1 answer
124
124 views
GO Classes DPP | GATE DA | Probability | Bayes Theorem
A company receives emails from three sources: $40\%$ are promotional emails, $35\%$ are personal emails, and $25\%$ are work emails. A filter marks an email as suspicious...
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1 1 vote
1 1 answer
133
133 views
GO Classes DPP | GATE CS | Probability | Independence of Events & Bernoulli Trials
A fair coin is tossed $8$ times and exactly $6$ heads are observed. The coin is tossed one more time under the same conditions. Given the previous result, what is the pro...
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1 1 vote
1 1 answer
116
116 views
GO Classes DPP | GATE CS | Probability | Bayes Theorem
A disease affects $2\%$ of a population. A medical test gives a positive result in $98\%$ of infected cases and in $3\%$ of non-infected cases. If a person is selected at...
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4 4 votes
3 3 answers
229
229 views
GO Classes DPP | GATE CS | Probability | Conditional Probability & Inclusion-Exclusion Principle
Assume $P(A)=0.4$, $P(B)=0.5$, and $P(A \cup B)=0.7$. Also, $P(C)=0.6$, $P(B \cap C) = 0.25$, and and $P(C \mid A \cap B)=\frac{1}{2}$. Find $P(A \mid B \cap C)$.$\frac{2...
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4 4 votes
1 1 answer
142
142 views
GO Classes DPP | GATE CS | Probability | Conditional Probability
In a probability space, it is given that $P(A)=a$, $P(B)=b$, and $P(B \mid A^c)=e$. Also, $P(A \mid B^c)=f$. Determine $P(A \mid B)$ and $P(B \mid A)$ in terms of $a$, $b...
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1 1 vote
1 1 answer
133
133 views
GO Classes DPP | GATE CS | Probability | Conditional Probability
Let $E$ and $F$ be two independent events with $P(E)=\frac{1}{3}$ and $P(F)=\frac{1}{2}$. Let $G$ be an event such that $G \subseteq E'$, $P(G)=\frac{1}{4}$, and $G$ is i...
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3 3 votes
2 2 answers
257
257 views
GO Classes DPP | GATE CS | Probability | Probability with Independent Trials
Ten tickets numbered $1$ to $10$ are sampled one at a time with replacement. The sampling is repeated $7$ times. Find the probability that the second largest number among...
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2 2 votes
1 1 answer
134
134 views
GO Classes DPP | GATE CS | Probability | Conditional Probability
A college cricket squad has $11$ players consisting of $4$ batsmen, $3$ all-rounders, $3$ bowlers and $1$ wicket keeper. Four players are selected randomly. Given that th...
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0 0 votes
1 1 answer
153
153 views
GO Classes DPP | GATE CS | Probability | Probability with Independent Trials
A bag contains $10$ blue marbles, $20$ green marbles, $30$ red marbles and $40$ yellow marbles. A marble is drawn from the bag, its colour is recorded, and then it is rep...
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1 1 vote
1 1 answer
140
140 views
GO Classes DPP | GATE CS | Probability | Inclusion-Exclusion Principle
A sample space has three events, $A$, $B$, and $C$. You are given the following probabilities regarding these events:$P(A') = 1/2$ $P(B') = 2/3$ $P(C) = 1/4$You are also ...
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2 2 votes
1 1 answer
126
126 views
GO Classes DPP | GATE DA | Probability | Conditional Probability
Let $A$ and $B$ be two events in a sample space such that $P(A' \cap B')=\frac{1}{6}$, $P(A \mid B)=\frac{3}{4}$, and $P(B \mid A')=\frac{2}{5}$. Find $P(A)$.$\frac{11}{1...
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1 1 vote
2 2 answers
170
170 views
GO Classes DPP | GATE DA | Probability | Binomial Distribution
For each element in a set of size $2n$, an unbiased coin is tossed independently. An element is chosen if the corresponding coin toss results in a head. What is the proba...
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3 3 votes
1 1 answer
123
123 views
GO Classes DPP | GATE DA | Probability | Conditional Probability
Let $A$ and $B$ be two events in a sample space such that $P(A \cup B) = 1$. Given $P(A) = p$ and $P(B) = q$ (where $q 0$), what is the value of $P(A^c \mid B)$ in terms...
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