Advertisements
Advertisements
प्रश्न
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
Advertisements
उत्तर
The repeated guessing of correct answers from multiple-choice questions is Bernoulli trials. Let X represent the number of correct answers by guessing in the set of 5 multiple-choice questions.
Probability of getting a correct answer is, p = `1/3`
` therefore q = 1 - p = 1 -1/3 = 2/3`
Clearly, X has a binomial distribution with n = 5 and p = `1/3`.
The p.m.f. of X is given by
P(X = x) = `"^nC_x p^x q^(n - x)`, x = 0, 1, 2, 4, 5
i.e. p(x) = `"^nC_x (1/3)^x (2/3)^(5-x)` x = 0, 1, 2, 3, 4, 5
P(four or more correct answers) = P[X ≥ 4] = p(4) + p(5)
`= ""^5C_4 (1/3)^4 (2/3)^(5 - 4) + "^5C_5 (1/3)^5 (2/3)^(5 - 5)`
`= 5xx(1/3)^4 xx (2/3)^1 + 1xx (1/3)^5 (2/3)^0`
`= (1/3)^4 [5 xx 2/3 + 1/3]`
`= (1/3)^4 [10/3 +1/3] = 1/81 xx 11/3 = 11/243`
Hence, the probability of getting four or more correct answers `11/243`.
संबंधित प्रश्न
A couple has two children, Find the probability that both children are females, if it is known that the elder child is a female.
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
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.
In a large bulk of items, 5 percent of the items are defective. What is the probability that a sample of 10 items will include not more than one defective item?
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.
The mathematics department has 8 graduate assistants who are assigned to the same office. Each assistant is just as likely to study at home as in office. How many desks must there be in the office so that each assistant has a desk at least 90% of the time?
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 at most 3 will survive .
It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that none contract the disease .
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.
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?
From a lot of 30 bulbs that includes 6 defective bulbs, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
The probability of a man hitting a target is 0.25. He shoots 7 times. What is the probability of his hitting at least twice?
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 none of the bulbs is defective .
Can the mean of a binomial distribution be less than its variance?
Determine the binomial distribution whose mean is 9 and variance 9/4.
Determine the binomial distribution whose mean is 20 and variance 16.
Find the binomial distribution whose mean is 5 and variance \[\frac{10}{3} .\]
The probability that an item produced by a factory is defective is 0.02. A shipment of 10,000 items is sent to its warehouse. Find the expected number of defective items and the standard deviation.
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).
A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.
The mean of a binomial distribution is 10 and its standard deviation is 2; write the value of q.
If in a binomial distribution n = 4 and P (X = 0) = \[\frac{16}{81}\] , find q.
If the mean and variance of a binomial distribution are 4 and 3, respectively, find the probability of no success.
In a box containing 100 bulbs, 10 are defective. What is the probability that out of a sample of 5 bulbs, none is defective?
A fair coin is tossed 100 times. The probability of getting tails an odd number of times 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
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
Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that none is a spade ?
Bernoulli distribution is a particular case of binomial distribution if n = ______
For Bernoulli Distribution, state formula for E(X) and V(X).
Explain why the experiment of tossing a coin three times is said to have binomial distribution.
Which one is not a requirement of a binomial distribution?
Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If P(x = r)/P(x = n – r) is independent of n and r, then p equals ______.
The sum of n terms of the series `1 + 2(1 + 1/n) + 3(1 + 1/n)^2 + ...` is
If x4 occurs in the tth term in the expansion of `(x^4 + 1/x^3)^15`, then the value oft is equal to:
If the coefficients of x7 and x8 in `(2 + x/3)^n` are equal, then n is
If a fair coin is tossed 10 times. Find the probability of getting at most six heads.
A box B1 contains 1 white ball and 3 red balls. Another box B2 contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes B1 and B2, then find the probability that the two balls drawn are of the same colour.
If a random variable X follows the Binomial distribution B(5, p) such that P(X = 0) = P(X = 1), then `(P(X = 2))/(P(X = 3))` 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.
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?
