Exercise 31.4 Archives - India Site

Exercise 31.4

Let $A$ and $B$ be two independent events such that $P(A)=D_{1}$ and $P(B)=p_{2}$. Describe in words the events whose probabilities are $p_{1}+p_{2}=2 p_{1} p_{2}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Events are said to be independent, if the occurrence or non – occurrence of one does not affect the probability of the occurrence or non – occurrence of the other.

Let $A$ and $B$ be two independent events such that $P(A)=D_{1}$ and $P(B)=p_{2}$. Describe in words the events whose probabilities are $1-\left(1-p_{1}\right)\left(1-p_{2}\right)$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Events are said to be independent, if the occurrence or non – occurrence of one does not affect the probability of the occurrence or non – occurrence of the other.

Let $A$ and $B$ be two independent events such that $P(A)=D_{1}$ and $P(B)=p_{2}$. Describe in words the events whose probabilities are (i) $p_{1} p_{2}$ (ii) $\left(1-p_{1}\right) p_{2}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Events are said to be independent, if the occurrence or non - occurrence of one does not affect the probability of the occurrence or non - occurrence of the other.

Two dice are thrown together and the total score is noted. The event $E_{1} F$ and $G$ are “a total $4$”, “a total of $9$ or more” and “a total divisible by $5^{\text {” }}$, respectively. Calculate $\mathrm{P}(\mathrm{E}), \mathrm{P}(\mathrm{F})$ and $\mathrm{P}(\mathrm{G})$ and decide which pairs of events, if any, are independent.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

The probabilities of two students $\mathrm{A}$ and $\mathrm{B}$ coming to the school in time are $\frac{3}{7}$ and $\frac{5}{7}$ respectively. Assuming that the events, ‘A coming in time’ and ‘B coming in time’ are independent, find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that the events 'A coming in time' and 'B coming in time' are independent. The advantage of coming to school in time is that you will not miss any part of the lecture and will be able to learn...

An urn contains $4\;red\;and\;7\;black\;balls.$ Two balls are drawn at random with replacement. Find the probability of getting one red and one blue ball.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

An urn contains $4\;red\;and\;7\;black\;balls.$ Two balls are drawn at random with replacement. Find the probability of getting $\begin{array}{ll}\text { (i) } 2 \text { red balls } & \text { (ii) } 2 \text { blue balls }\end{array}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that one of them is black and other is red

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that (i) Both balls are red (ii) First ball is black and second is red.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A die is thrown thrice. Find the probability of getting an odd number at least once.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

The odds against a certain event are $5$ to $2$ and the odds in favour of another event, independent to the former are 6 to 5 . Find the probability that (a) at least one of the events will occur, and (b) none of the events will occur.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

An anti-aircraft gun can take a maximum of $4\;shots$ at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are $0.4,0.3,0.2$ and $0.1$ respectively. What is the probability that the gun hits the plane?

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

The probability that $A$ hits a target is $\frac{1}{3}$ and the probability that $B$ hits it, is $\frac{2}{5}$ What is the probability that the target will be hit, if each one of $A$ and $B$ shoots at the target?

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

An article manufactured by a company consists of two parts $X$ and $Y$. In the process of manufacture of the part $x, 9$ out of $100$ parts may be defective. Similarly, $5$ out of $100$ are likely to be defective in the manufacture of part $Y$. Calculate the probability that the assembled product will not be defective.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Three cards are drawn with replacement from a well shuffled pack of cards. Find the probability that the cards drawn are king, queen and jack.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A bag contains $3\;red$ and $2\;black$ balls. One ball is drawn from it at random. Its color is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing first red and second black ball.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A bag contains $3\;red$ and $2\;black$ balls. One ball is drawn from it at random. Its color is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing (i) two red balls, (ii) two black balls

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

An unbiased die is tossed twice. Find the probability of getting $4,\;5,\;or\;6$ on the first toss and $1,2,3$ or 4 on the second toss.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Given the probability that $A$ can solve a problem is $\frac{2}{3}$ and the probability that $B$ can solve the same problem is $\frac{3}{5}$. Find the probability that none of the two will be able to solve the problem.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A die is tossed twice. Find the probability of getting a number greater than $3$ on each toss.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

If $A$ and $B$ are two independent events such that $P(A \cup B)=0.60$ and $P(A)=0.2$ find $P(B)$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

$A$ and $B$ are two independent events. The probability that $A$ and $B$ occur is $\frac{1}{6}$ and the probability that neither of them oocurs is $\frac{1}{3}$. Find the probability of occurrence of two events.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question, ......(i)

If $A$ and $B$ are two independent events such that $p(\bar{A} \cap B)=\frac{2}{15}$ and $P(A \cap \bar{B})=\frac{1}{6}$, then find $P(B)$.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

If $P(not B)=0.65, P(A \cup B)=0.85$, and $A$ and $B$ are independent events, then find $P(A)$.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Given two independent events $A$ and $B$ such that $P(A)=0.3$ and $P(B)=0.6$, Find $P\left(\frac{B}{A}\right)$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$

Given two independent events $A$ and $B$ such that $P(A)=0.3$ and $P(B)=0.6$, Find (i) $P(A \cup B)$ (ii) $P\left(\frac{A}{B}\right) \quad$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$ (i) (ii)

Given two independent events $A$ and $B$ such that $P(A)=0.3$ and $P(B)=0.6$, Find $ \text { (i) } P(\bar{A} \cap B) \text { (ii) } P(\bar{A} \cap \bar{B}) $

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$ (i) (ii)

Given two independent events $A$ and $B$ such that $P(A)=0.3$ and $P(B)=0.6$, Find $ \text { (i) } P(A \cap B) \text { (ii) } P(A \cap \bar{B}) $

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$

If $A$ and $B$ be two events such that $P(A)=\frac{1}{4}, P(B)=\frac{1}{3}$ and $P(A \cup B)=\frac{1}{2}$, show that $A$ and $B$ are independent events.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A coin is tossed three times. Let the events $A, B$ and $C$ be defined as follows: $A=$ first toss is head, $B=$ second toss is head, and $C=$ exactly two heads are tossed in a row. $\begin{array}{lll}\text { Check the independence of } \; C \text { and } A\end{array}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A coin is tossed three times. Let the events $A, B$ and $C$ be defined as follows: $A=$ first toss is head, $B=$ second toss is head, and $C=$ exactly two heads are tossed in a row. $\begin{array}{lll}\text { Check the independence of (i) } A \text { and } B & \text { (ii) } B \text { and } C \end{array}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A card is drawn from a pack of $52\;cards$ so that each card is equally likely to be selected. State whether events $A$ and $B$ are independent if, $A=$ the card drawn is spade, $B=$ the card drawn in an ace

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

(i) A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. State whether events $A$ and $B$ are independent if, $A=$ the card drawn is a king or queen $B=$ the card drawn is a queen or jack (ii) A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. State whether events $A$ and $B$ are independent if, $A=$ the card drawn is black, $B=$ the card drawn is a king

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question, (i) (ii)

Prove that in throwing a pair of dice, the occurrence of the number $4$ on the first die is independent of the occurrence of $5$ on the second die.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A coin is tossed thrice and all the eight outcomes are assumed equally likely. State whether events $A$ and $B$ are independent if, $A=$ the number of heads is two, $B=$ the last throw results in head

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

(i) A coin is tossed thrice and all the eight outcomes are assumed equally likely. State whether events $A$ and $B$ are independent if, $A=$ the first throw results in head, $B=$ the last throw results in tail (ii) A coin is tossed thrice and all the eight outcomes are assumed equally likely. State whether events $A$ and $B$ are independent if, $A=$ the number of heads is odd, $B=$ the number of tails is odd

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question, So, $A\;and\;B$ are independent events. (ii)