Have you ever wondered how these weather apps predict the chance of rain or how games of chance like dice or cards work? Probability has an answer to it! Probability is the branch of mathematics that helps us understand and measure uncertainty in everyday life, whether in predicting the weather, outcomes in a game or chances in real-life decisions. It uses logic, data and formulas to calculate how likely an event is to occur. This exercise contains questions from a series of concepts like sample space, types of events, conditional probability etc.
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The NCERT Solutions for Chapter 14 Miscellaneous Exercise are designed to help you in understanding the concepts by offering detailed explanations and step-wise calculations. The NCERT solutions will help you sharpen your reasoning and analytical thinking. Follow the NCERT notes page to learn more!
Answer:
Given,
No. of red marbles = 10
No. of blue marbles = 20
No. of green marbles = 30
Total number of marbles = 10 + 20 + 30 = 60
Number of ways of drawing 5 marbles from 60 marbles = $^{60}\textrm{C}_{5}$
(i) .
Number of ways of drawing 5 blue marbles from 20 blue marbles = $^{20}\textrm{C}_{5}$
$\therefore$ Probability of drawing all blue marbles = $\frac{^{20}\textrm{C}_{5}}{^{60}\textrm{C}_{5}}$
Answer:
Given,
No. of red marbles = 10
No. of blue marbles = 20
No. of green marbles = 30
Total number of marbles = 10 + 20 + 30 = 60
Number of ways of drawing 5 marbles from 60 marbles = $^{60}\textrm{C}_{5}$
(ii). We know,
The probability that at least one marble is green = 1 - Probability that no marble is green
Now, Number of ways of drawing no green marbles = Number of ways of drawing only red and blue marbles
= $^{(20+10)}\textrm{C}_{5} = ^{30}\textrm{C}_{5}$
$\therefore$ The probability that no marble is green = $\frac{^{30}\textrm{C}_{5}}{^{60}\textrm{C}_{5}}$
$\therefore$ The probability that at least one marble is green = $1-\frac{^{30}\textrm{C}_{5}}{^{60}\textrm{C}_{5}}$
Answer:
Total number of ways of drawing 4 cards from a deck of 52 cards = $^{52}\textrm{C}_{4}$
We know that there are 13 diamonds and 13 spades in a deck.
Now, Number of ways of drawing 3 diamonds and 1 spade = $^{13}\textrm{C}_{3}.^{13}\textrm{C}_{1}$
$\therefore$ Probability of obtaining 3 diamonds and 1 spade
$= \frac{^{13}\textrm{C}_{3}.^{13}\textrm{C}_{1}}{^{52}\textrm{C}_{4}}$
Question 3: (i) A die has two faces each with number '1', three faces each with number ‘2’ and one face with number ‘3’. If die is rolled once, determine
Answer:
Total number of faces of a die = 6
(i) Number of faces with number '2' = 3
$P(2) = \frac{3}{6}$ $=\frac{1}{2}$
Therefore, required probability P(2) is 0.5
Question 3: (ii) A die has two faces each with number ‘1’, three faces each with number ‘2’ and one face with number ‘3’. If die is rolled once, determine
P(1 or 3)
Answer:
Total number of faces of a die = 6
(ii) P (1 or 3) = P (not 2) = 1 − P (2)
$= 1-P(2) = 1-\frac{3}{6}$ $=\frac{1}{2}$
Therefore, required probability P(1 or 3) is 0.5
Question 3: (iii) A die has two faces each with number ‘1’, three faces each with number ‘2’ and one face with number ‘3’. If die is rolled once, determine
P(not 3)
Answer:
Total number of faces of a die = 6
(iii) Number of faces with number '3' = 1
$\therefore$ $P(3) = \frac{1}{6}$
$\therefore$ P(not 3) = 1 - P(3)
$= 1-\frac{1}{6} = \frac{5}{6}$
Therefore, required probability P(not 3) is $\frac{5}{6}$
Answer:
Given, Total number of tickets sold = 10,000
Number of prizes awarded = 10
(a) If one ticket is bought,
P(getting a prize) = $\frac{10}{10000} = \frac{1}{1000}$
$\therefore$ P(not getting a prize) = 1 - P(getting a prize)
$1- \frac{1}{1000} = \frac{999}{1000}$
Answer:
Given, Total number of tickets sold = 10,000
Number of prizes awarded = 10
(b) If two tickets are bought:
Number of tickets not awarded = 10000 - 10 = 9990
$\therefore$ P(not getting a prize) = $\frac{^{9990}\textrm{C}_{2}}{^{10000}\textrm{C}_{2}}$
Answer:
Given, Total number of tickets sold = 10,000
Number of prizes awarded = 10
(c) If ten tickets are bought:
Number of tickets not awarded = 10000 - 10 = 9990
$\therefore$ P(not getting a prize) = $\frac{^{9990}\textrm{C}_{10}}{^{10000}\textrm{C}_{10}}$
you both enter the same section?
Answer:
Total number of students = 100
Let A and B be the two sections consisting of 40 and 60 students respectively.
Number of ways of selecting 2 students out of 100 students.= $^{100}\textrm{C}_{n}$
If both are in section A:
Number of ways of selecting 40 students out of 100 = $^{100}\textrm{C}_{40}$ (The remaining 60 will automatically be in section B!)
Remaining 38 students are to be chosen out of (100-2 =) 98 students
$\therefore$ Required probability if they both are in section A = $\frac{^{98}\textrm{C}_{38}}{^{100}\textrm{C}_{40}}$
Similarly,
If both are in section B:
Number of ways of selecting 60 students out of 100 = $= ^{100}\textrm{C}_{60} = ^{100}\textrm{C}_{40}$ (The remaining 40 will automatically be in section A!)
Remaining 58 students are to be chosen out of (100-2 =) 98 students
$\therefore$ Required probability if they both are in section B = $\frac{^{98}\textrm{C}_{58}}{^{100}\textrm{C}_{60}}$
Required probability that both are in same section = Probability that both are in section A + Probability that both are in section B
= $\frac{^{98}\textrm{C}_{38}}{^{100}\textrm{C}_{40}}+\frac{^{98}\textrm{C}_{58}}{^{100}\textrm{C}_{60}}$
$\\ = \frac{^{98}\textrm{C}_{38}+^{98}\textrm{C}_{58}}{^{100}\textrm{C}_{40}} \\ \\ =\frac{\frac{98!}{38!.60!} + \frac{98!}{58!.40!}}{\frac{100!}{40!.60!}}$
$=\frac{85}{165} = \frac{17}{33}$
Hence, the required probability that both are in same section is $\frac{17}{33}$
you both enter the different sections?
Answer:
Total number of students = 100
Let A and B be the two sections consisting of 40 and 60 students respectively.
We found out in (a) that the probability that both students are in same section is $\frac{17}{33}$
(b) Probability that both the students are in different section = $1 - \frac{17}{33}$ $= \frac{16}{33}$
Answer:
Given, 3 letters are put in 3 envelopes.
The number of ways of putting the 3 different letters randomly = 3!
Number of ways that at least one of the 3 letters is in the correct envelope
= No. of ways that exactly 1 letter is in correct envelope + No. of ways that 2 letters are in the correct envelope(The third is automatically placed correctly)
= No. of ways that exactly 1 letter is in correct envelope + No. of ways that all the 3 letters are in the correct envelope
= $(^{3}\textrm{C}_{1}\times1) + 1 = 4$
(Explanation for $^{3}\textrm{C}_{1}\times1$ :
No. of ways of selecting 1 envelope out of 3 = $^{3}\textrm{C}_{1}$.
If we put the correct letter in it, there is only one way the other two are put in the wrong envelope! )
Therefore, the probability that at least one letter is in its proper envelope = $\frac{4}{3!} = \frac{4}{6} = \frac{2}{3}$
Question 7: (i) A and B are two events such that $\small P(A)=0.54$ , $\small P(B)=0.69$ and $\small P(A\cap B)=0.35$ Find $\small P(A\cup B)$
Answer:
Given, P(A) = 0.54, P(B) = 0.69, P(A $\cap$ B) = 0.35
(i) We know, P(A $\cup$ B) = P(A) + P(B) - P(A $\cap$ B)
$\implies$ P(A $\cup$ B) = 0.54 + 0.69 - 0.35 = 0.88
$\implies$ P(A $\cup$ B) = 0.88
Question 7: (ii) A and B are two events such that $\small P(A)=0.54$, $\small P(B)=0.69$ and $\small P({A}\cap {B})=0.35$. Find $\small P({A}'\cap {B}')$
Answer:
Given, $P(A) = 0.54, P(B) = 0.69, P(A \cap B) = 0.35$
And, $P(A \cup B) = 0.88$
(ii) $A' \cap B'= (A \cup B)'$ [De Morgan’s law]
So, $P(A' \cap B') = P(A \cup B)' = 1 - P(A \cup B) = 1 - 0.88 = 0.12$
$\therefore$ $P(A' \cap B') = 0.12$
Question 7: (iii) A and B are two events such that $\small P(A)=0.54$ , $\small P(B)=0.69$ and $\small P(A\cap B)=0.35.$ Find $\small P(A\cap B{}')$
Answer:
Given, P(A) = 0.54, P(B) = 0.69, P(A $\cap$ B) = 0.35
And, P(A $\cup$ B) = 0.88
(iii) P(A $\cap$ B') = P(A-B) = P(A) - P(A $\cap$ B)
= 0.54 - 0.35 = 0.19
$\therefore$ P(A $\cap$ B') = 0.19
Question 7: (iv) A and B are two events such that $\small P(A)=0.54$, $\small P(B)=0.69$ and $\small P(A\cap B)=0.35$ . Find $\small P(B\cap A{}')$
Answer:
Given, P(A) = 0.54, P(B) = 0.69, P(A $\cap$ B) = 0.35
And, P(A $\cup$ B) = 0.88
(iv) P(B $\cap$ A') = P(B-A) = P(B) - P(A $\cap$ B)
= 0.69 - 0.35 = 0.34
$\therefore$ P(B $\cap$ A') = 0.34
S. No. | Name | Sex | Age in years |
1. | Harish | M | 30 |
2. | Rohan | M | 33 |
3. | Sheetal | F | 46 |
4. | Alis | F | 28 |
5. | Salim | M | 41 |
A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over $\small 35$ years?
Answer:
Given,
Total number of persons = 5
No. of male spokesperson = 3
No. of spokesperson who is over 35 years of age = 2
Let E be the event that the spokesperson is a male and F be the event that the spokesperson is over 35 years of age.
$\therefore$ $P(E) = \frac{3}{5}\ and\ P(F) = \frac{2}{5}$
Since only one male is over 35 years of age,
$\therefore$ $P(E\cap F) = \frac{1}{5}$
We know,
$P(E \cup F) = P(E) + P(F) - P(E \cap F)$
$\implies P(E \cup F) = \frac{3}{5} + \frac{2}{5} - \frac{1}{5} = \frac{4}{5}$
Therefore, the probability that the spokesperson will either be a male or over 35 years of age is $\frac{4}{5}$.
Answer:
Since 4-digit numbers greater than 5000 are to be formed,
The $1000's$ place digit can be filled up by either 7 or 5 in $^{2}\textrm{C}_{1}$ ways
Since repetition is allowed,
Each of the remaining 3 places can be filled by any of the digits 0, 1, 3, 5, or 7 in $5$ ways.
$\therefore$ Total number of 4-digit numbers greater than 5000 = $^{2}\textrm{C}_{1}\times5\times5\times5 -1$
$= 250 - 1 = 249$ (5000 cannot be counted, hence one less)
We know, a number is divisible by 5 if unit’s place digit is either 0 or 5.
$\therefore$ Total number of 4-digit numbers greater than 5000 that are divisible by 5 = $^{2}\textrm{C}_{1}\times5\times5\times^{2}\textrm{C}_{1} -1$ $= 100 - 1 = 99$
Therefore, the required probability =
$P(with\ repetition) = \frac{99}{249} = \frac{33}{83}$
Answer:
Since 4-digit numbers greater than 5000 are to be formed,
The $1000's$ place digit can be filled up by either 7 or 5 in $^{2}\textrm{C}_{1}$ ways
Since repetition is not allowed,
The remaining 3 places can be filled by remaining 4 digits in $^{4}\textrm{C}_{3}\times3!$ ways.
$\therefore$ Total number of 4-digit numbers greater than 5000 = $^{2}\textrm{C}_{1}\times(^{4}\textrm{C}_{3}\times3!)= 2\times4\times6 = 48$
We know, a number is divisible by 5 if unit’s place digit is either 0 or 5.
Case 1. When digit at $1000's$ place is 5, the units place can be filled only with 0.
And the $100's$ & $10's$ places can be filled with any two of the remaining digits {1,3,7} in $^{3}{C}_{2}\times2!$
$\therefore$ Number of 4-digit numbers starting with 5 and divisible by 5 = $1\times ^{3}{C}_{2}\times2! = 1.3.2 = 6$
Case 2. When digit at $1000's$ place is 7, the units place can be filled by 0 or 5 in 2 ways.
And the $100's$ & $10's$ places can be filled with any two of the remaining 3 digits in $^{3}{C}_{2}\times2!$
$\therefore$ Number of 4-digit numbers starting with 7 and divisible by 5 = $1\times2\times( ^{3}{C}_{2}\times2! )= 1.2.3.2 = 12$
$\therefore$ Total number of 4-digit numbers greater than 5000 that are divisible by 5 = 6 + 12 = 18
Therefore, the required probability =
$P(without\ repetition) = \frac{18}{48} = \frac{3}{8}$
Answer:
Given, Each wheel can be labelled with 10 digits.
Number of ways of selecting 4 different digits out of the 10 digits = $^{10}\textrm{C}_{4}$
These 4 digits can arranged among themselves is $4!$ ways.
$\therefore$ Number of four digit numbers without repetitions =
$^{10}\textrm{C}_{4}\times4! = \frac{10!}{4!.6!}\times4! = 10\times9\times8\times7 = 5040$
Number of combination that can open the suitcase = 1
$\therefore$ Required probability of getting the right sequence to open the suitcase = $\frac{1}{5040}$
Also read
1. Experiments and Outcomes
An experiment is considered random if it produces diverse results even when it is conducted again under the same circumstances. Each possible result is called an outcome.
2. Sample Space and Events
The sample space is the set of all possible outcomes of an experiment. An event is a subset of the sample space that we are interested in.
3. Classical Probability
If each outcome is equally likely then
$$
P(E)=\frac{\text { Number of favourable outcomes }}{\text { Total number of outcomes }}
$$
4. Addition Theorem of Probability
For any two events $A$ and $B$ -
$$
P(A \cup B)=P(A)+P(B)-P(A \cap B)
$$
5. Multiplication Theorem of Probability
For independent events-
$$
P(A \cap B)=P(A) \cdot P(B)
$$
For conditional probability-
$$
P(A \cap B)=P(A) \cdot P(B \mid A)
$$
6. Conditional Probability
It refers to the probability of an event occurring given that another event has already occurred.
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Frequently Asked Questions (FAQs)
Sample space this experiment = { HH, HT, TH, TT}
Probability of both head = 1/4
Sample space this experiment = { HH, HT, TH, TT}
Probability of at least one head = 3/4
Sample space = { 1, 2, 3, 4, 5, 6}
The probability of getting a number less than or equal to 3 = 1/2
Calculus has 5 marks weightage in the CBSE Class 11 Maths.
statistics and probability have 10 marks weightage in the CBSE Class 11 Maths.
Mathematical Reasoning has 2 marks weightage in the CBSE Class 11 Maths.
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