Advertisements
Advertisements
प्रश्न
It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective?
उत्तर
The repeated selections of articles in a random sample space are Bernoulli trails. Let X denote the number of times of selecting defective articles in a random sample space of 12 articles.
Clearly, X has a binomial distribution with n = 12 and p = 10% = `10/100 = 1/10`
q = 1 - p = `1 - 1/10 = 9/10`
Given: n = 12
∴ X ~ B `(12, 1/10)`
The p.m.f. of X is given by
P[X = x] = `"^nC_x p^x q^(n - x)`
i.e. p(x) = `"^12C_x (1/10)^x (9/10)^(12 - x)`, x = 1, 2, 3,...,12
P(9 defective articles) = P[X = 9]
= p(9) = `"^12C_9 (1/10)^9 (9/10)^(12 - 9)`
`= (12!)/(9! 3!) (1/10)^9 (9/10)^3`
`= (12 xx 11 xx 10 xx 9!)/(9! xx 3 xx 2 xx 1) xx 1/10^9 xx 9^3/10^3`
`= 2 xx 11 xx 10 * 9^3/10^12 = 22 (9^3/10^11)`
Hence, the probability of getting 9 defective articles `=22 (9^3/10^11)`.
संबंधित प्रश्न
The probability that a bomb will hit a target is 0.8. Find the probability that out of 10 bombs dropped, exactly 4 will hit the target.
Given X ~ B (n, p)
If n = 10 and p = 0.4, find E(X) and var (X).
There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?
Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that
- all the five cards are spades?
- only 3 cards are spades?
- none is a spade?
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. What is the probability that out of 5 such bulbs
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one, will fuse after 150 days of use.
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?
In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true’; if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly.
Suppose X has a binomial distribution `B(6, 1/2)`. Show that X = 3 is the most likely outcome.
(Hint: P(X = 3) is the maximum among all P (xi), xi = 0, 1, 2, 3, 4, 5, 6)
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is 1/100. What is the probability that he will in a prize (a) at least once (b) exactly once (c) at least twice?
Find the probability of getting 5 exactly twice in 7 throws of a die.
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
(A) 10−1
(B) `(1/2)^5`
(C) `(9/10)^5`
(D) 9/10
The probability that a student is not a swimmer is 1/5 . Then the probability that out of five students, four are swimmers is
(A) `""^5C_4 (4/5)^4 1/5`
(B) `(4/5)^4 1/5
(C) `""^5C_1 1/5 (4/5)^4 `
(D) None of these
How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%?
A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times.
Five cards are drawn one by one, with replacement, from a well-shuffled deck of 52 cards. Find the probability that
(i) all the five cards diamonds
(ii) only 3 cards are diamonds
(iii) none is a diamond
The probability of a man hitting a target is 1/4. If he fires 7 times, what is the probability of his hitting the target at least twice?
If getting 5 or 6 in a throw of an unbiased die is a success and the random variable X denotes the number of successes in six throws of the die, find P (X ≥ 4).
Eight coins are thrown simultaneously. Find the chance of obtaining at least six heads.
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?
A bag contains 2 white, 3 red and 4 blue balls. Two balls are drawn at random from the bag. If X denotes the number of white balls among the two balls drawn, describe the probability distribution of X.
An urn contains four white and three red balls. Find the probability distribution of the number of red balls in three draws with replacement from the urn.
Find the probability distribution of the number of sixes in three tosses of a die.
Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the mean and variance of number of red cards.
A coin is tossed 5 times. If X is the number of heads observed, find the probability distribution of X.
A man wins a rupee for head and loses a rupee for tail when a coin is tossed. Suppose that he tosses once and quits if he wins but tries once more if he loses on the first toss. Find the probability distribution of the number of rupees the man wins.
An unbiased coin is tossed 8 times. Find, by using binomial distribution, the probability of getting at least 6 heads.
Suppose that a radio tube inserted into a certain type of set has probability 0.2 of functioning more than 500 hours. If we test 4 tubes at random what is the probability that exactly three of these tubes function for more than 500 hours?
The probability that a certain kind of component will survive a given shock test is \[\frac{3}{4} .\] Find the probability that among 5 components tested exactly 2 will survive .
Assume that the probability that a bomb dropped from an aeroplane will strike a certain target is 0.2. If 6 bombs are dropped, find the probability that exactly 2 will strike the target .
The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university only one will graduate .
The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university all will graduate .
In a 20-question true-false examination, suppose a student tosses a fair coin to determine his answer to each question. For every head, he answers 'true' and for every tail, he answers 'false'. Find the probability that he answers at least 12 questions correctly.
Suppose X has a binomial distribution with n = 6 and \[p = \frac{1}{2} .\] Show that X = 3 is the most likely outcome.
In a multiple-choice examination with three possible answers for each of the five questions out of which only one is correct, what is the probability that a candidate would get four or more correct answers just by guessing?
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \[\frac{1}{100} .\] What is the probability that he will win a prize exactly once.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \[\frac{1}{100} .\] What is the probability that he will win a prize at least twice.
The probability of a shooter hitting a target is \[\frac{3}{4} .\] How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99?
How many times must a man toss a fair coin so that the probability of having at least one head is more than 80% ?
A factory produces bulbs. The probability that one bulb is defective is \[\frac{1}{50}\] and they are packed in boxes of 10. From a single box, find the probability that exactly two bulbs are defective
A factory produces bulbs. The probability that one bulb is defective is \[\frac{1}{50}\] and they are packed in boxes of 10. From a single box, find the probability that more than 8 bulbs work properly
Can the mean of a binomial distribution be less than its variance?
Determine the binomial distribution whose mean is 9 and variance 9/4.
If the mean and variance of a binomial distribution are respectively 9 and 6, find the distribution.
Find the binomial distribution when the sum of its mean and variance for 5 trials is 4.8.
If the probability of a defective bolt is 0.1, find the (i) mean and (ii) standard deviation for the distribution of bolts in a total of 400 bolts.
If on an average 9 ships out of 10 arrive safely at ports, find the mean and S.D. of the ships returning safely out of a total of 500 ships.
In eight throws of a die, 5 or 6 is considered a success. Find the mean number of successes and the standard deviation.
Find the expected number of boys in a family with 8 children, assuming the sex distribution to be equally probable.
A dice is thrown thrice. A success is 1 or 6 in a throw. Find the mean and variance of the number of successes.
If a random variable X follows a binomial distribution with mean 3 and variance 3/2, find P (X ≤ 5).
The mean and variance of a binomial distribution are \[\frac{4}{3}\] and \[\frac{8}{9}\] respectively. Find P (X ≥ 1).
If the sum of the mean and variance of a binomial distribution for 6 trials is \[\frac{10}{3},\] find the distribution.
A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.
A die is thrown three times. Let X be 'the number of twos seen'. Find the expectation of X.
In a binomial distribution, if n = 20 and q = 0.75, then write its mean.
If in a binomial distribution mean is 5 and variance is 4, write the number of trials.
If the mean of a binomial distribution is 20 and its standard deviation is 4, find p.
The mean of a binomial distribution is 10 and its standard deviation is 2; write the value of q.
If the mean and variance of a binomial variate X are 2 and 1 respectively, find P (X > 1).
If the mean and variance of a binomial distribution are 4 and 3, respectively, find the probability of no success.
If in a binomial distribution n = 4, P (X = 0) = \[\frac{16}{81}\], then P (X = 4) equals
A rifleman is firing at a distant target and has only 10% chance of hitting it. The least number of rounds he must fire in order to have more than 50% chance of hitting it at least once is
A fair coin is tossed a fixed number of times. If the probability of getting seven heads is equal to that of getting nine heads, the probability of getting two heads is
A fair coin is tossed 100 times. The probability of getting tails an odd number of times is
A fair die is thrown twenty times. The probability that on the tenth throw the fourth six appears is
If X is a binomial variate with parameters n and p, where 0 < p < 1 such that \[\frac{P\left( X = r \right)}{P\left( X = n - r \right)}\text{ is } \] independent of n and r, then p equals
Let X denote the number of times heads occur in n tosses of a fair coin. If P (X = 4), P (X= 5) and P (X = 6) are in AP, the value of n is
One hundred identical coins, each with probability p of showing heads are tossed once. If 0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, the value of p is
If the mean and variance of a binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than 1 is
A biased coin with probability p, 0 < p < 1, of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is 2/5, then p equals
If X follows a binomial distribution with parameters n = 8 and p = 1/2, then P (|X − 4| ≤ 2) equals
In a binomial distribution, the probability of getting success is 1/4 and standard deviation is 3. Then, its mean is
A coin is tossed 4 times. The probability that at least one head turns up is
A coin is tossed n times. The probability of getting at least once is greater than 0.8. Then, the least value of n, is
Mark the correct alternative in the following question:
A box contains 100 pens of which 10 are defective. What is the probability that out of a sample of 5 pens drawn one by one with replacement at most one is defective?
Mark the correct alternative in the following question:
The probability that a person is not a swimmer is 0.3. The probability that out of 5 persons 4 are swimmers is
A bag contains 7 red, 5 white and 8 black balls. If four balls are drawn one by one with replacement, what is the probability that all are white ?
A bag contains 7 red, 5 white and 8 black balls. If four balls are drawn one by one with replacement, what is the probability that any two are white ?
Determine the binomial distribution where mean is 9 and standard deviation is `3/2` Also, find the probability of obtaining at most one success.
Find the mean and variance of the random variable X which denotes the number of doublets in four throws of a pair of dice.
Bernoulli distribution is a particular case of binomial distribution if n = ______
Which one is not a requirement of a binomial distribution?
If X follows binomial distribution with parameters n = 5, p and P(X = 2) = 9, P(X = 3), then p = ______.
The mean, median and mode for binomial distribution will be equal when
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is:-
A pair of dice is thrown four times. If getting a doublet is considered a success then find the probability of two success.
If a fair coin is tossed 10 times. Find the probability of getting at most six heads.
An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is ______.
The mean and variance of a binomial distribution are α and `α/3` respectively. If P(X = 1) = `4/243`, then P(X = 4 or 5) is equal to ______.
If a random variable X follows the Binomial distribution B (33, p) such that 3P(X = 0) = P(X = 1), then the value of `(P(X = 15))/(P(X = 18)) - (P(X = 16))/(P(X = 17))` is equal to ______.
The probability of hitting a target in any shot is 0.2. If 5 shots are fired, find the probability that the target will be hit at least twice.
In three throws with a pair of dice find the chance of throwing doublets at least twice.
A student is given a quiz with 10 true or false questions and he answers by sheer guessing. If X is the number of questions answered correctly write the p.m.f. of X. If the student passes the quiz by getting 7 or more correct answers what is the probability that the student passes the quiz?
If the sum of mean and variance of a binomial distribution is `25/9` for 5 trials, find p.
