Intersection of Sets

Q.

Sean and Evan are college roommates who have part-time jobs as servers in restaurants.

The distribution of Sean’s weekly income is approximately normal with mean $\$225$ and standard deviation $\$25$.

The distribution of Evan’s weekly income is approximately normal with mean $\$240$ and standard deviation $\$15$.

Assuming their weekly incomes are independent of each other, which of the following is closest to the probability that Sean will have a greater income than Evan in a randomly selected week?

1. $0.67$

2. $0.7000$

3. $0.227$

4. $0.303$

5. $0.354$

Q.

One card is drawn at random from a pack of 52 cards. What's the probability that it is a king or queen ?

Q. 1 card is drawn from a well-shuffled deck of 52 card. Calculate the probability that the card well be not an ace

Q.

Let $f:R\to R$ be a function which satisfies $f(x+y)=f\left(x\right)+f\left(y\right)\forall x,y\in R$. If $f\left(1\right)=2$ and $g\left(n\right)=\underset{k=1}{\overset{n-1}{\sum f\left(k\right)}}$, $n\in N$ then the value of $n$ for which $g\left(n\right)=20$is:

1. $9$

2. $5$

3. $4$

4. $20$

Q.

The probability that a man can hit a target is $\frac{3}{4}$. He tries $5$times. The probability that he will hit the target at least three times is

1. $\frac{291}{364}$

2. $\frac{371}{464}$

3. $\frac{471}{502}$

4. $\frac{459}{512}$

Q.

Find the probability distribution of

number of heads in four tosses of a coin.

Q.

If $A$ and $B$ are two independent events, then $P\left(\frac{A}{B}\right)=$

1. $0$

2. $1$

3. $P\left(A\right)$

4. $P\left(B\right)$

Q.

Let $A$ and $B$ be two non-empty subsets of a set $X$ such that $A$is not a subset of $B$, then

1. $A$is always a number of the complement of $B$

2. $B$ is always a subset of $A$

3. $A$and $B$ are always disjoint

4. $A$ and the complement of $B$ are always non–disjoint

Q.

In an examination, $80\%$ of the students passed in English, $85\%$ in Mathematics and$75\%$ in both English and Mathematics. If $40$students failed in both the subjects, find the total number of students.

1. $350$

2. $400$

3. $450$

4. $500$

Q.

The set $A=\mathrm{\{}x:x$ belongs to $\mathrm{ℝ},x^2=16$ and $2x=6\}$ is equal to

1. $\varnothing$

2. $\left\{14,3,4\right\}$

3. $\left\{3\right\}$

4. $\left\{4\right\}$

Q.

If a coin be tossed $n$ times then probability that the head comes odd times is

1. $\frac{1}{2}$

2. $\frac{1}{2^n}$

3. $\frac{1}{2^{n-1}}$

4. None of these

Q.

The total number of numbers, lying between $100$ and $1000$ that can be formed with the digits $1,2,3,4,5$, if the repetition of digits is not allowed and numbers are divisible by either $3$ or $5$ is

Q. A survey of people in a given region showed that $20\%$ were smokers. The probability of death due to lung cancer, given that a person smoked, was $10$ times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to lung cancer in the region is $0.006$, what is the probability of death due to lung cancer given that a person is a smoker?

1. $\frac{1}{140}$
2. $\frac{1}{70}$
3. $\frac{1}{10}$
4. $\frac{3}{140}$

Q.

If A and B are mutually exclusive events such that P(A = 0.35 and P(B=0.45, find

(i) $P(A\cup B)$

(ii) $P(A\cap B)$

(iii) $P(A\cap \overline{B})$

(iv) $P(\overline{A}\cap \overline{B})$

Q.

The lengths of human pregnancies are normally distributed with a mean of $268\text{days}$ and a standard deviation of $15\text{days}$.

What is the probability that a pregnancy lasts at least $300\text{days}$?

1. $0.4834$

2. $0.0166$

3. $0.0332$

4. $0.9834$

5. $0.0179$

Q.

In a binomial distribution the probability of getting a success is $\frac{1}{4}$ and standard deviation is $3$, then its mean is

1. $6$

2. $8$

3. $10$

4. $12$

Q.

One die is thrown three times and the sum of the thrown numbers is $15$. The probability for which number $4$ appears in first throw

1. $\frac{1}{18}$

2. $\frac{1}{36}$

3. $\frac{1}{9}$

4. $\frac{1}{3}$

Q.

A particular telephone number is used to receive both voice

calls and fax messages.

Suppose that $25\%$ of the incoming calls involve fax messages, and consider a sample of $25$ incoming calls.

What is the probability that

$a$. At most $6$ of the calls involve a fax message?

$b$. Exactly $6$ of the calls involve a fax message?

$c$. At least $6$ of the calls involve a fax message?

$d$. More than $6$ of the calls involve a fax message?

$e$. What is the expected number of calls among the $25$ that involve a fax message?

$f$. What is the standard deviation of the number among the $25$ calls that involve a fax message?

$g$. What is the probability that the number of calls among the $25$ that involve a fax transmission exceeds the expected number by more than $2$ standard deviations?

Q.

If $A$ and $B$ are two independent events such that $P\left(A\right)=\frac{1}{2}$ and $P\left(B\right)=\frac{1}{3}$, then $P$(neither $A$ nor $B$) is equal to

1. $\frac{2}{3}$

2. $\frac{1}{6}$

3. $\frac{5}{6}$

4. $\frac{1}{3}$

Q. Five horses are in a race. Mr. A selects two of the horses at random and belts on them. The probability that Mr. A selected the winning horse is

1. 
2. 
3. 
4. 

Q.

The probability that a missile hits a target successfully is $0.75.$ In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than $0.95$, is

Q.

If $A$ and $B$ are two independent events such that $P\left(B\right)=\frac{2}{7}$, $P(A\cup B)=0.8$, then $P\left(A\right)$ is equal to

1. $0.1$

2. $0.2$

3. $0.3$

4. $0.4$

Q.

In binomial probability distribution, the mean is $3$ and the standard deviation is $\frac{3}{2}.$ Then the probability distribution is:

1. ${\left(\frac{3}{4}+\frac{1}{4}\right)}^{12}$

2. ${\left(\frac{1}{4}+\frac{3}{4}\right)}^{12}$

3. ${\left(\frac{1}{4}+\frac{3}{4}\right)}^9$

4. ${\left(\frac{3}{4}+\frac{1}{4}\right)}^9$

Q. Of the students in a school, it is known that $30$$\%$ have
$100$$\%$ attendance and $70\%$ students are irregular. Previous year results report that $70\%$ of all students who have $100\%$ attendance attain $A$ grade and $10\%$ irregular students attain $A$ grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an $A$ grade. What is the probability that the student has $100\%$ attendance ? Is regularity required only in school ? Justify your answer.

Q. A room has three lamps .From a collectionof 10 light bulbs of which 6 are no good, a person selects 3 at random and puts the in socket.What is the probability that he will have the light?

Q.

A survey shows that $61\%,46\%$, and $29\%$ of the people watched “$3$ idiots”, “Rajneeti” and “Avatar” respectively. $25\%$ of people watched exactly two of the three movies and $3\%$ watched none. What percentage of people watched all three movies ?

1. $39\%$

2. $11\%$

3. $14$

4. $7\%$

Q.

$12$ defective pens are accidentally mixed with $132$ good ones$.$ It is not possible to just look at a pen and tell whether or not it is defective$.$ One pen is taken out at random from this lot$.$ Determine the probability that the pen taken out is a good one$.$

Q.

The probability that a student will pass the final examination in both

English and Hindi is 0.5 and the probability of passing neither is 0.1. If the

probability of passing the English examination is 0.75. What is the

probability of passing the Hindi examination ?

Q.

The probability of choosing at random a number that is divisible by $6$ or $8$ from among $1$ to $90$ is equal to

1. $\frac{1}{6}$

2. $\frac{1}{30}$

3. $\frac{11}{80}$

4. $\frac{23}{90}$

Q.

In a lottery $50$ tickets are sold in which $14$ are prizes. A man bought $2$ tickets, then the probability that the man win the prize is

1. $\frac{17}{35}$

2. $\frac{18}{35}$

3. $\frac{72}{175}$

4. $\frac{13}{175}$