Find the best tutors and institutes for Class 12 Tuition
Q4:
Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?
Solution :
Step 1: Identifying the Distribution Model
The problem involves a fixed number of independent Bernoulli trials, where each trial (selecting a person) has two possible outcomes: right-handed (success) or not right-handed (failure). Since the probability of success is constant, we apply the Binomial Distribution formula:
$P(X = k) = \binom{n}{k} p^k q^{n-k}$
Where:
- $n = 10$ (Total number of people in the sample)
- $p = 0.9$ (Probability that a person is right-handed)
- $q = 1 - p = 0.1$ (Probability that a person is not right-handed)
- $X$ is the random variable representing the number of right-handed people.
Step 2: Defining the Target Probability
We are required to find the probability that at most 6 people are right-handed. This is expressed as:
$P(X \leq 6) = \sum_{k=0}^{6} \binom{10}{k} (0.9)^k (0.1)^{10-k}$
Due to the law of total probability, it is computationally more efficient to calculate the complement and subtract from 1:
$P(X \leq 6) = 1 - [P(X=7) + P(X=8) + P(X=9) + P(X=10)]$
Step 3: Calculating Individual Probabilities
We compute each term $P(X=k) = \binom{10}{k} (0.9)^k (0.1)^{10-k}$:
| $k$ | Calculation: $\binom{10}{k} (0.9)^k (0.1)^{10-k}$ | Value |
|---|---|---|
| $7$ | $\frac{10!}{7!3!} (0.9)^7 (0.1)^3 = 120 \times 0.4782969 \times 0.001$ | $0.0573956$ |
| $8$ | $\frac{10!}{8!2!} (0.9)^8 (0.1)^2 = 45 \times 0.4304672 \times 0.01$ | $0.1937102$ |
| $9$ | $\frac{10!}{9!1!} (0.9)^9 (0.1)^1 = 10 \times 0.3874205 \times 0.1$ | $0.3874205$ |
| $10$ | $\frac{10!}{10!0!} (0.9)^{10} (0.1)^0 = 1 \times 0.3486784 \times 1$ | $0.3486784$ |
Step 4: Final Summation and Subtraction
Summing the probabilities for $k > 6$:
$\sum_{k=7}^{10} P(X=k) = 0.0573956 + 0.1937102 + 0.3874205 + 0.3486784 = 0.9872047$
Subtracting from total probability [Per the Complement Rule]:
$P(X \leq 6) = 1 - 0.9872047 = 0.0127953$
Visual Representation of Distribution
Final Solution: The probability that at most 6 people are right-handed is approximately 0.0128.
More Questions from Class 12 Mathematics Probability Miscellaneous Exercise on Chapter 13
- Q1(i): $A$ and $B$ are two events such that $P(A) \neq 0$. Find $P(B|A)$, if (i) $A$ is a subset of $B$
- Q1(ii): $A$ and $B$ are two events such that $P(A) \neq 0$. Find $P(B|A)$, if (ii) $A \cap B = \phi$
- Q2(i): A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male.
- Q2(ii): A couple has two children, (ii) Find the probability that both children are females, if it is known that the elder child is a female.
- Q3: Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.
- Q5: If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?
- Q6: Suppose we have four boxes A,B,C and D containing coloured marbles as given below: One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C?
- Q7: Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
- Q8: If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\frac{1}{2}$).
- Q9(i): An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known: $P(A \text{ fails}) = 0.2$, $P(B \text{ fails alone}) = 0.15$, $P(A \text{ and } B \text{ fail}) = 0.15$. Evaluate the following probabilities (i) $P(A \text{ fails|B has failed})$
- Q9(ii): An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known: $P(A \text{ fails}) = 0.2$, $P(B \text{ fails alone}) = 0.15$, $P(A \text{ and } B \text{ fail}) = 0.15$. Evaluate the following probabilities (ii) $P(A \text{ fails alone})$
- Q10: Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.
- Q11: If A and B are two events such that $P(A) \neq 0$ and $P(B | A) = 1$, then
- Q12: If $P(A|B) > P(A)$, then which of the following is correct :
- Q13: If A and B are any two events such that $P(A) + P(B) – P(A \text{ and } B) = P(A)$, then
CBSE Solutions for Class 12 Mathematics Probability
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Probability
Top Tutor
Karampura, Delhi
10 yrs of
Exp
1 student
2500
per hour
Online Classes
Student's Home
I teach and help students to achieve immediate and long term goals. My experience as a personal tutor and the founder of an online educational startup 'MinDely' has helped him gain knowledge of child psychology and understanding students needs and takes measures accordingly. I plan the lessons as per the students ability to cope up and moreover upon their needs. My teaching technique is student friendly and easy to understand which helps students understand it better. Weekly test conducted by him helps the parents to know the students performance.
Top Tutor
Koramangala 8th Block, Bangalore
8 yrs of
Exp
15 students
55 hrs of classes completed
700
per hour
Online Classes
I believe mathematics becomes easy once it is learned the right way. Currently pursuing a PhD in Mathematics with 7 years of teaching experience for JEE Main & Advanced, CBSE, AS/A Levels, IB HL/SL, and IGCSE, State Boards. I focus on conceptual understanding and problem-solving rather than rote learning. My goal is to help students think mathematically, build confidence, and truly enjoy the subject with patience and clarity.
Great teacher he's great at explaining topics, he is fast at teaching so you should ask all doubts immediately he gives lots of homework but still recommended.
Top Tutor
KR Puram, Bangalore
8 yrs of
Exp
2 students
31 hrs of classes completed
500
per hour
Online Classes
I am a physics teacher handling X,XI,XII grades, having 8 years experience as Science HOD in a cbse School. I have a degree in BE ( electronics and telecommunications).With my best efforts, support, patience ,commitment I unlock my students potential.Concepts are presented in a clear, simple and direct manner. Consistently my class result is 100% with good subject average. Expanding pupils horizons is my accomplishment.
Top Tutor
Rajarajeshwari Nagar, Bangalore
12 yrs of
Exp
51 students
1960 hrs of classes completed
500
per hour
Online Classes
Student's Home
I have been training students for 12th Board examinations for more than a decade now.
Top Tutor
Tagore Garden, Delhi
5 yrs of
Exp
15 students
326 hrs of classes completed
500
per hour
Online Classes
I,Mridula Das from Delhi ama professional Psychology teacher. I coach XI and XII grade CBSE students. I holds a Masters degree in Psychology and a B.Ed degree, and have also completed my MBA in Human Resources Management. Further ,I hold a PG Diploma in Guidance and Counselling. I have been teaching for the past 3 years in schools in West Delhi, and have guided 25-30 students per batch. According to my teaching style, I keep my class interactive and responsive and incorporate several visual aids in teaching. As a psychological expert, I understands student's minds set better and charters her teaching style accordingly. I teaches ,both online and offline classes as well.
Mridula ma'am is an excellent psychology teacher. What I loved most about her teaching style is that she always used really relatable and easy to understand examples to explain concepts, which made even the difficult topics feel simple. Her dedication and patience truly make her stand out as a teacher.I was able to score well, thanks to her guidance.
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Probability Miscellaneous Exercise on Chapter 13 worksheets
Download Now
