Advertisements
Advertisements
प्रश्न
Solve the following problem :
It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on at most 2 days of a week.
उत्तर
Let X denote the number of days it rains in a week.
P(it rains) = p = `(10)/(30) = (1)/(3)`
∴ q = 1 – p = `1 - (1)/(3) = (2)/(3)`
Given, n = 7
∴ X ~ B`(7, 1/3)`
The p.m.f. of X is given by
P(X = x) = `""^7"C"_x (1/3)^x (2/3)^(7 - x),x` = 0, 1, ...,7
P(it rains at most 2 days of a week)
= P(X ≤ 2)
= P(X = 0 or X = 1 or X = 2)
= P(X = 0) + P(X = 1) + P(X = 2)
= `""^7"C"_0(1/3)^0 (2/3)^7 + ""^7"C"__1 (1/3)(2/3)^6 + ""^7"C"_2(1/3)^2 (2/3)^5`
= `(2^5)/(3^7)(4 + 14 + (7 xx 6 xx 5!)/(2 xx 1 xx 5!))`
= `(2^5)/(3^7)(18 + 21)`
= `39(2^5/3^7)`.
APPEARS IN
संबंधित प्रश्न
Probability distribution of X is given by
| X = x | 1 | 2 | 3 | 4 |
| P(X = x) | 0.1 | 0.3 | 0.4 | 0.2 |
Find P(X ≥ 2) and obtain cumulative distribution function of X
State the following are not the probability distributions of a random variable. Give reasons for your answer.
| Y | -1 | 0 | 1 |
| P(Y) | 0.6 | 0.1 | 0.2 |
Find the probability distribution of number of heads in two tosses of a coin.
Find the probability distribution of number of heads in four tosses of a coin.
If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)19 = 0⋅1348]
Which of the following distributions of probabilities of a random variable X are the probability distributions?
(i)
| X : | 3 | 2 | 1 | 0 | −1 |
| P (X) : | 0.3 | 0.2 | 0.4 | 0.1 | 0.05 |
| X : | 0 | 1 | 2 |
| P (X) : | 0.6 | 0.4 | 0.2 |
(iii)
| X : | 0 | 1 | 2 | 3 | 4 |
| P (X) : | 0.1 | 0.5 | 0.2 | 0.1 | 0.1 |
(iv)
| X : | 0 | 1 | 2 | 3 |
| P (X) : | 0.3 | 0.2 | 0.4 | 0.1 |
Find the probability distribution of the number of heads, when three coins are tossed.
Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
An urn contains 4 red and 3 blue balls. Find the probability distribution of the number of blue balls in a random draw of 3 balls with replacement.
Find the mean and standard deviation of each of the following probability distribution :
| xi : | -2 | -1 | 0 | 1 | 2 |
| pi : | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |
A discrete random variable X has the probability distribution given below:
| X: | 0.5 | 1 | 1.5 | 2 |
| P(X): | k | k2 | 2k2 | k |
Determine the mean of the distribution.
A fair die is tossed. Let X denote twice the number appearing. Find the probability distribution, mean and variance of X.
A random variable has the following probability distribution:
| X = xi : | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P (X = xi) : | 0 | 2 p | 2 p | 3 p | p2 | 2 p2 | 7 p2 | 2 p |
The value of p is
If X is a random-variable with probability distribution as given below:
| X = xi : | 0 | 1 | 2 | 3 |
| P (X = xi) : | k | 3 k | 3 k | k |
The value of k and its variance are
Mark the correct alternative in the following question:
For the following probability distribution:
| X: | −4 | −3 | −2 | −1 | 0 |
| P(X): | 0.1 | 0.2 | 0.3 | 0.2 | 0.2 |
The value of E(X) is
A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.
Two fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. Also find E(X).
Verify the following function, which can be regarded as p.m.f. for the given values of X :
| X = x | -1 | 0 | 1 |
| P(x) | -0.2 | 1 | 0.2 |
John and Mathew started a business with their capitals in the ratio 8 : 5. After 8 months, john added 25% of his earlier capital as further investment. At the same time, Mathew withdrew 20% of bis earlier capital. At the end of the year, they earned ₹ 52000 as profit. How should they divide the profit between them?
The following table gives the age of the husbands and of the wives :
| Age of wives (in years) |
Age of husbands (in years) |
|||
| 20-30 | 30- 40 | 40- 50 | 50- 60 | |
| 15-25 | 5 | 9 | 3 | - |
| 25-35 | - | 10 | 25 | 2 |
| 35-45 | - | 1 | 12 | 2 |
| 45-55 | - | - | 4 | 16 |
| 55-65 | - | - | - | 4 |
Find the marginal frequency distribution of the age of husbands.
From the following data, find the crude death rates (C.D.R.) for Town I and Town II, and comment on the results :
| Age Group (in years) | Town I | Town II | ||
| Population | No. of deaths | Population | No. of deaths | |
| 0-10 | 1500 | 45 | 6000 | 150 |
| 10-25 | 5000 | 30 | 6000 | 40 |
| 25 - 45 | 3000 | 15 | 5000 | 20 |
| 45 & above | 500 | 22 | 3000 | 54 |
An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X?
A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes
The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of X = 0
Solve the following problem :
If a fair coin is tossed 4 times, find the probability that it shows 3 heads
Solve the following problem :
If a fair coin is tossed 4 times, find the probability that it shows head in the first 2 tosses and tail in last 2 tosses.
Solve the following problem :
The probability that a lamp in the classroom will burn is 0.3. 3 lamps are fitted in the classroom. The classroom is unusable if the number of lamps burning in it is less than 2. Find the probability that the classroom cannot be used on a random occasion.
Solve the following problem :
The probability that a component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 components tested survive.
A random variable X has the following probability distribution
| X | 2 | 3 | 4 |
| P(x) | 0.3 | 0.4 | 0.3 |
Then the variance of this distribution is
Find the probability distribution of the number of doublets in three throws of a pair of dice
Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X and variance of X
Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1), "for" x = 1"," 2"," 3"," 4),(2"k"x, "for" x = 5"," 6"," 7),(0, "Otherwise"):}`
where k is a constant. Calculate the value of k
A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the box. Find the probability distribution of the number of unspoiled fruits. Also find the mean of the probability distribution.
Five numbers x1, x2, x3, x4, x5 are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order such that x1 < x2 < x3 < x4 < x5. What is the probability that x2 = 7 and x4 = 11?
