Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let X = number of fuse bulbs.
p = probability of a bulb produced by a factory will fuse after 150 days of use.
∴ p = 0.05 and q = 1 – p = 1 – 0.05 = 0.95
X has a binomial distribution with n = 5 and p = 0.05
∴ X ~ B(5, 0.05)
The p.m.f. of X is given by
P(X = x) = nCx px qn−x
i.e., p(x) = 5Cx (0.05)x (0.95)5−x, x = 0, 1, 2, 3, 4, 5
(i) P(none) = P[X = 0]
= p(0)
= 5C0 (0.05)0 (0.95)5 - 0
= 1 × 1 × (0.95)5
= (0.95)5
Hence, the probability that none of the bulbs will fuse after 150 days = (0.95)5.
(ii) P(not more than one) = P(X ≤ 1)
= p(0) + p(1)
= 5C0 (0.05)0 (0.95)5 - 0 + 5C1 (0.05)1 (0.95)5 - 1
= 5C0 (0.05)0 (0.95)5 + 5C1 (0.05)1 (0.95)4
= 1 × 1 × (0.95)5 + 5 × (0.05) × (0.95)4
= (0.95)4 [0.95 + 5(0.05)]
= (0.95)4 [0.95 + 0.25]
= (0.95)4 (1.20)
= (1.2)(0.95)4
Hence, the probability that not more than one bulb will fuse after 150 days = (1.2)(0.95)4
(iii) P (more than 1) = P(X > 1)
= 1 - P[X ≤ 1]
= 1 - (1.2)(0.95)4
Hence, the probability that more than one bulb fuse after 150 days = 1 – (1.2)(0.95)4.
(iv) P (at least one) = P(X ≥ 1)
= 1 - P[X = 0]
= 1 - p(0)
= 1 - 5C0 (0.05)0 (0.95)5 - 0
= 1 - 1 × 1 × (0.95)5
= 1 - (0.95)5
Hence, the probability that at least one bulb fuses after 150 days = 1 - (0.95)5.
संबंधित प्रश्न
Given that X ~ B(n= 10, p). If E(X) = 8 then the value of
p is ...........
(a) 0.6
(b) 0.7
(c) 0.8
(d) 0.4
A fair coin is tossed 8 times. Find the probability that it shows heads at least once
A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes.
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?
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?
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?
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
A couple has two children, Find the probability that both children are males, if it is known that at least one of the children is male.
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?
A coin is tossed 5 times. If X is the number of heads observed, find the probability distribution of X.
The items produced by a company contain 10% defective items. Show that the probability of getting 2 defective items in a sample of 8 items is
\[\frac{28 \times 9^6}{{10}^8} .\]
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 .
In a hospital, there are 20 kidney dialysis machines and the chance of any one of them to be out of service during a day is 0.02. Determine the probability that exactly 3 machines will be out of service on the same day.
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.
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.
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.
In eight throws of a die, 5 or 6 is considered a success. Find the mean number of successes and the standard deviation.
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 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.
If the mean of a binomial distribution is 20 and its standard deviation is 4, find p.
If the mean and variance of a binomial distribution are 4 and 3, respectively, find the probability of no success.
If for a binomial distribution P (X = 1) = P (X = 2) = α, write P (X = 4) in terms of α.
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 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
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
If X follows a binomial distribution with parameters n = 100 and p = 1/3, then P (X = r) is maximum when r =
A fair die is tossed eight times. The probability that a third six is observed in the eighth throw is
If the mean and variance of a binomial distribution are 4 and 3, respectively, the probability of getting exactly six successes in this distribution 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:
Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If \[\frac{P\left( X = r \right)}{P\left( X = n - r \right)}\] is independent of n and r, then p equals
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
Mark the correct alternative in the following question:
The probability of guessing correctly at least 8 out of 10 answers of a true false type examination 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 ?
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 a fair coin is tossed 10 times. Find the probability of getting at most six heads.
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(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?
A fair coin is tossed 6 times. Find the probability of getting heads 4 times.
