JEE Maths
Probability
Binomial Distribution:
A random variable X taking values 0, 1, 2, ..., n is said to have a binomial distribution
with parameters n and p, if its probability distribution is given by
P (X = r) = ncr pr qn–r,
where q = 1 – p and r = 0, 1, 2, ..., n.
Example: If a coin is tossed 10 times, find the probability of exactly 5 heads.
Solution: let X denote the number of heads.
So X has the binomial distribution with n = 10 an p = ½
∴ P(X = x) = nCx qn-x px , x = 1,2,3....,n
∴ P(X = x) = 10cx(1/2)10-x(1/2)x = 10cx(1/2)10.
so P(X = 6) = 10c5(1/2)10= (10 !)/ (5 !×5!) ×( 1/210) = 63/ 256
Example:A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
Solution:
Let X denote the number of balls marked with the digit 0 among the 4 balls drawn.
Since the balls are drawn with replacement, the trials are Bernoulli trials.
X has a binomial distribution with n = 4 and P= 1/10
∴ Q= 1-P =1− 1/10 = 9/10
∴ P(X = x) = nCx qn-x px , x = 1,2,3....,n
= 4Cx ( 9/10 )n−x( 1/10 )x
P (none marked with 0) = P (X = 0)
= 4C0 ( 9/10 )4 ( 1/10 )0
= 1×( 9/10)4
= ( 9/10 )4
Example: Find the probability of getting 5 exactly twice in 7 throws of a dice.
Solution:
The repeated tossing of a dice are Bernoulli trials. Let X represent the number of times of getting 5 in 7 throws of the dice.
Probability of getting 5 in a single throw of the dice is P=16
∴Q=1-P= 1− 1/6 = 5/6
Clearly, X has the probability distribution with n = 7 and P=16
∴P(X = x) = nCx qn-x px
= 7Cx (5/6)7−x (1/6)x
P (getting 5 exactly twice) = P(X = 2)
=7C2 (5/6)5 (1/6)2
= 21×(5/6)5× 1/36
= 7 /12 ×(5/6)5
JEE Maths Probability NCERT Chapter 16 Free Notes for Best Revision
Revision of JEE Maths Probability is a crucial aspect of effective learning. Revision plays a vital role in the learning process and is especially important before exams. Here are some key points you can consider emphasizing in your content:
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- Spacing Effect: Spacing out revision sessions over time, rather than cramming all at once, has been shown to improve long-term retention. Create a revision schedule leading up to the exams to allow for spaced practice.
- Regular Revision over Cramming: Regular and consistent revision throughout the academic year is very important. Waiting until the last moment to cram everything can be overwhelming and less effective than spaced-out revision.
- Self-Assessment: Assess your understanding periodically through quizzes or self-tests. This helps you to gauge your progress and identify areas that need further attention. Refer JEE Probability Online Tests.
- Balanced Approach: Remind students to strike a balance between revision and other activities. Adequate rest, exercise, and relaxation are essential for optimal learning and performance.
- Seeking Help: If you face difficulties during the revision process, Refer the videos of JEE Maths Probability. Clearing doubts promptly is crucial for a better grasp of the subject matter. You can always ask your doubts on Probability JEE Maths NCERT Chapter 16. “Ask a Question” section of LearnoHub.com
By highlighting the benefits and strategies of effective revision, you can approach your studies more mindfully and achieve better results in your exams. Best of luck bachhon!
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