probability formulae

 Let S be the sample space and A a subset of S

P(A)=n(A)/n(S)

 

P(S)=1

P( {} )  = 0,  {} stands for null set

 

let A' denote the event that "A does not happen"

P(A) +P(A') =1

 

P(A')  = 1-P(A)

P(A) = 1-P(A')

 

addition theorem

P[A ∪ B] = P(A) +P(B) - P[A∩B] 

P[A or B] = P(A) +P(B) - P[A and B] 

 

Bayes theorem

Total probability

 

 

problem 2
A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

solution to problem 2 of  bayes theorem

problem 3
Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostelier?

solution to problem 3 of  bayes theorem


problem 4
In answering a question on a multiple choice test, a student either knows the answer or guesses. Let ( 3/4 ) be the probability that he knows the answer and ( 1/4) be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability ( 1/4 ). What is the probability that the student knows the answer given that he answered it correctly?
solution to bayes theorem problem 4 for ncert cbse 12th mathematics

problem 5

A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive ?
solution to bayes theorem problem 5 for ncert cbse 12th mathematics

problem 6

There are three coins. One is a two headed coin (having head on both faces),another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?

solution to problem 6 of bayes theorem for cbse ncert mathematics

problem 7

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

solution to problem 7 of bayes theorem for cbse ncert mathematics 

problem 8
 A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further,2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?
solution of problem 8 on bayes theorem for cbse mathematics 

problem 9
Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.
solution of problem 9 on bayes theorem for cbse mathematics

problem 10
Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?
solution of problem10 on bayes theorem for ncert cbse mathematics

problem 11
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
solution of problem11 on bayes theorem for ncert cbse mathematics

problem 12
A card from a pack of 52 cards is lost. From the remaining cards of the pack,two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
 solution to bayes theorem problem 12 for ncert cbse mathematics probability

problem 13
Probability that a man speaks truth is (  4 / 5 ). A coin is tossed and the man reports that a head appeared.Find the probability that actually there was a head.
 solution to bayes theorem problem 13for ncert cbse mathematics probability


miscellaneous exercise problem 14
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 a 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?
solution to miscellaneous exercise problem 14 on bayes theorem in  ncert cbse 12th mathematics

miscellaneous exercise problem 15
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.
solution to miscellaneous exercise problem 15on bayes theorem in  ncert cbse 12th mathematics


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bayes theorem is a topic for cbse / ncert  / scert of 12th standard in India. These are some of the important questions from the topic of bayes theorem from various cbse ncert textbooks and old question paper