Union of Sets

Q. Suppose an integer from 1 through 1000 is chosen at random, find the probability that the integer is a multiple of 2 or a multiple of 9.

Q.

The probability of happening at least one of the events $A$ and $B$ is $0.6$. If the events $A$and $B$ happens simultaneously with the probability $0.2$, then $P\left(A^{}\right)+P\left(B^{}\right)=$

1. $0.4$

2. $0.8$

3. $1.4$

4. $1.2$

Q.

Thirty days are in September, April, June and November. Some months are of thirty one days. A month is chosen at random. Then its probability of having exactly three days less than a maximum of $31$ day is

1. $\frac{15}{16}$

2. $1$

3. $\frac{3}{48}$

4. None of these

Q.

Two cards are drawn without replacement from a well shuffled pack. Find the probability that one of them is an ace of heart

1. $\frac{1}{25}$

2. $\frac{1}{26}$

3. $\frac{1}{52}$

4. None of these

Q.

Let $A,B,C$ be three events. If the probability of occurring exactly one event out of $A$ and $B$ is $1-a$, out of $B$ and $C$ and $A$ is $1-2a$ and that of occurring three events simultaneously is $a^2$, then the probability that at least one out of $A,B,C$ will occur is

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

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

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

4. Greater than $\frac{3}{4}$

Q.

One card is drawn from a pack of 52 cards. The probability that it is the card of a king or spade is

1. $\frac{1}{26}$

2. $\frac{3}{26}$

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

4. $\frac{3}{13}$

Q. Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that $A$ or $B$ occurs is $\frac{1}{2}$, then the probability of both of them occur together is :

1. $0.10$
2. $0.02$
3. $0.20$
4. $0.01$

Q.

Find the probability of drawing an ace or a spade from a deck of $52$ cards in a single draw?

Q. One number is chosen from numbers 1 to 100. Find the probability that it is divisible by 4 or 6?

Q. Probability of solving a particular question by person $A$ is $\frac{1}{3}$ and probability of solving that question by person $B$ is $\frac{2}{5}.$ What is the probability of solving that question by at least one of them ?

1. $\frac{2}{5}$
2. $\frac{2}{3}$
3. $\frac{3}{5}$
4. $\frac{7}{9}$

Q. There are $3$ bags, each containing $5$ white balls and $3$ black balls. Also there are $2$ bags, each containing $2$ white balls and $4$ black balls. A white ball is drawn at random. If probability that this white ball is from a bag of the first group is $k$, then the value of $61k-40$ is

Q.

A die is thrown twice. What is the probability that at least one of the two throws come up with the number 3?

Q. All face cards from pack of $52$ playing cards are removed. From remaining $40$ cards, two are drawn randomly, without replacement, then probability of drawing a pair (same denominations) is

1. $\frac{1}{13}$
2. $\frac{1}{78}$
3. $\frac{2}{39}$
4. $\frac{4}{13}$

Q.

A card is drawn from a deck of 52 cards. Find the probability of getting an ace or a spade card.

Q.

In an assembly of $4$ persons the probability that at least $2$ of them have the same birthday, is:

1. $0.293$

2. $0.24$

3. $0.0001$

4. $0.016$

Q.

A candidate takes three tests in succession and the probability of passing the first test is $p$. The probability of passing each succeeding test is $p$ or $\frac{p}{2}$ according as he passes or fails in the preceding one. The candidate is selected, if he passes atleast two tests. The probability that the candidate is selected, is

1. $p^2(2–p)$

2. $p(2–p)$

3. $p+p^2+p^3$

4. $p^2(1–p)$

Q.

If $\text{A}$ and $\text{B}$ are two events of a random experiment, such that $\text{P}\left(\text{A∪B}\right)\text{=}\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5,\text{P}\left(\text{A∪B}\right)\text{=}{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$10,$}\right.}$}\right.$ and $\text{P(B) =}\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\text{,}$ then the value of $\text{P(A)}$ equals

1. $\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$10$}\right.$

2. $\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$10$}\right.$

3. $\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$10$}\right.$

4. $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$

Q. If the probability of a six-digit number $N$ whose six digits are $1,2,3,4,5,6$ when written in random order, is divisible by $6$ is $p,$ then the value of $\frac{1}{p}$ is

Q. From a deck of $52$ cards, two cards are missing. One card is drawn at random from the remaining cards and found to be a spade. Find the probability that
$\left(1\right)$ both the missing cards were spades.
$\left(2\right)$ one of the missing card was spade.
$\left(3\right)$ neither of the missing cards was spade.
$\left(4\right)$ both the missing cards were same suit.

Q. Miss $X$ takes either tea or coffee at morning break. If she had tea one morning, the probability that she has tea the next morning is $\mathrm{0.4.}$ If she had coffee one morning, the probability that she has coffee the next morning is $\mathrm{0.3.}$ Suppose she has coffee on a Monday morning. The probability that she has tea on the following Wednesday morning, is

1. $0.46$
2. $0.51$
3. $0.61$
4. $0.49$

Q.

A die is thrown. If $\mathrm{A}$ is the event that the number obtained is greater than $3$ and $\mathrm{B}$ is the event that the number obtained is less than$5$. Then, $\mathrm{P}(\mathrm{A}\cup \mathrm{B})$ is

1. $\frac{2}{5}$

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

3. $0$

4. $1$

Q.

From past experience it is known that an investor will invest in security $A$ with a probability of $0.6$, will invest in security $B$ with a probability $0.3$and will invest in both $A$ and $B$with a probability of $0.2$. What is the probability that an investor will invest neither in $A$ nor in $B$?

1. $0.7$

2. $0.28$

3. $0.3$

4. $0.4$

Q.

In a non-leap year, the probability of having $53$ Fridays or Saturdays is:

1. $\frac{3}{7}$

2. $\frac{4}{7}$

3. $\frac{2}{7}$

4. $\frac{1}{7}$

Q. If $A$ and $B$ are two events of a random experiment such that $P\left(A/B\right)=P\left(B/A\right)\ne 0$, then which of the following is/are correct:

1. $A\subset B$ but $A\ne B$
2. $n\left(A\right)=n\left(B\right)$
3. $A\cap B=\varphi$
4. $P\left(A\right)=P\left(B\right)$

Q. Given that $E$ and $F$ are events such that $P\left(E\right)=0.6,P\left(F\right)=0.3$ and $P(E\cap F)=0.2$, then the value of $P\left(E/F\right)$ and $P\left(F/E\right)$ is:

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

Q.

Let $A$, $B$ and $C$ be the three events such that $P\left(A\right)=0.3$, $P\left(B\right)=0.4$, $P\left(C\right)=0.8$, $P(A\cap B)=0.08$, $P(A\cap C)=0.28$ and $P(A\cap B\cap C)=0.09$. If $P(A\cup B\cup C)\ge 0.75$, then $P(B\cap C)$satisfies

1. $P(B\cap C)\le 0.23$

2. $P(B\cap C)\le 0.48$

3. $0.23\le P(B\cap C)\le 0.48$

4. $0.23\le P(B\cap C)\ge 0.48$

Q.

A card is drawn from a pack of $52$ cards. A gambler bets that it is a spade or an ace. What are the odds against his winning this bet

1. $17:52$

2. $52:17$

3. $9:4$

4. $4:9$

Q.

Find the probability distribution of

number of heads in two tosses of a coin

number of tails in the simultaneous tosses of three coins

number of heads in four tosses of a coin.

Q.

If A and B are two events associated with a random experiment such that $P\left(A\right)=0.3,P\left(B\right)=0.4$ and $P(A\cup B)$ = 0.5, find $P(A\cap B)$

Q.

The probability of happening of two events A and B are 0.25 and 0.50 respectively. If the probability of happening of A and B together is 0.14, then probability that neither A nor B happens is

1. 0.39

2. 0.25

3. 0.11

4. none of these