Advertisements
Advertisements
प्रश्न
Solve the following problem :
Find the probability of the number of successes in two tosses of a die, where success is defined as six appears in at least one toss.
उत्तर
Success is defined as a number six appears on at least one die.
Let X denote the number of successes.
∴ possible values of X are 0, 1, 2.
Let P(getting six) = p =`(1)/(6)`
∴ q = 1 – p = `1 - (1)/(6) = (5)/(6)`
∴ P(X = 0) = P(no success) = qq = q2 = `(25)/(36)`
P(X =1) = P(one success) pq + qp = 2pq
= `2 xx (1)/(6) xx (5)/(6)`
= `(10)/(36)`
P(X = 2) = P(two successes) = pp = p2 = `(1)/(36)`
∴ Probability distribution of X is as follows:
| X | 0 | 1 | 2 |
| P(X = x) | `(25)/(36)` | `(10)/(36)` | `(1)/(36)` |
APPEARS IN
संबंधित प्रश्न
An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represents the number of black balls. What are the possible values of X? Is X a random variable?
Find the probability distribution of number of tails in the simultaneous tosses of three coins.
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
There are 4 cards numbered 1 to 4, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X.
Five defective bolts are accidently mixed with twenty good ones. If four bolts are drawn at random from this lot, find the probability distribution of the number of defective bolts.
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 : | 1 | 3 | 4 | 5 |
| pi: | 0.4 | 0.1 | 0.2 | 0.3 |
Find the mean and standard deviation of each of the following probability distribution:
| xi : | −1 | 0 | 1 | 2 | 3 |
| pi : | 0.3 | 0.1 | 0.1 | 0.3 | 0.2 |
Find the mean and standard deviation of each of the following probability distribution :
| xi : | 1 | 2 | 3 | 4 |
| pi : | 0.4 | 0.3 | 0.2 | 0.1 |
A fair coin is tossed four times. Let X denote the number of heads occurring. Find the probability distribution, mean and variance of X.
A fair die is tossed. Let X denote 1 or 3 according as an odd or an even number appears. Find the probability distribution, mean and variance of X.
A fair coin is tossed four times. Let X denote the longest string of heads occurring. Find the probability distribution, mean and variance of X.
A box contains 13 bulbs, out of which 5 are defective. 3 bulbs are randomly drawn, one by one without replacement, from the box. Find the probability distribution of the number of defective bulbs.
Three cards are drawn at random (without replacement) from a well shuffled pack of 52 cards. Find the probability distribution of number of red cards. Hence, find the mean of the distribution .
An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X ? Is X a random variable ? If yes, then find the mean and variance of X.
Write the values of 'a' for which the following distribution of probabilities becomes a probability distribution:
| X= xi: | -2 | -1 | 0 | 1 |
| P(X= xi) : |
\[\frac{1 - a}{4}\]
|
\[\frac{1 + 2a}{4}\]
|
\[\frac{1 - 2a}{4}\]
|
\[\frac{1 + a}{4}\]
|
Find the mean of the following probability distribution:
| X= xi: | 1 | 2 | 3 |
| P(X= xi) : |
\[\frac{1}{4}\]
|
\[\frac{1}{8}\]
|
\[\frac{5}{8}\]
|
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes and, hence, find its mean.
Let X be a random variable which assumes values x1 , x2, x3 , x4 such that 2P (X = x1) = 3P (X = x2) = P (X = x3) = 5P (X = x4). Find the probability distribution of X.
Calculate `"e"_0^circ ,"e"_1^circ , "e"_2^circ` from the following:
| Age x | 0 | 1 | 2 |
| lx | 1000 | 880 | 876 |
| Tx | - | - | 3323 |
Find mean and standard deviation of the continuous random variable X whose p.d.f. is given by f(x) = 6x(1 - x);= (0); 0 < x < 1(otherwise)
A departmental store gives trafnfng to the salesmen in service followed by a test. It is experienced that the performance regarding sales of any salesman is linearly related to the scores secured by him. The following data gives the test scores and sales made by nine (9) salesmen during a fixed period.
| Test scores (X) | 16 | 22 | 28 | 24 | 29 | 25 | 16 | 23 | 24 |
| Sales (Y) (₹ in hundreds) | 35 | 42 | 57 | 40 | 54 | 51 | 34 | 47 | 45 |
(a) Obtain the line of regression of Y on X.
(b) Estimate Y when X = 17.
Three different aeroplanes are to be assigned to carry three cargo consignments with a view to maximize profit. The profit matrix (in lakhs of ₹) is as follows :
| Aeroplanes | Cargo consignments | ||
| C1 | C2 | C3 | |
| A1 | 1 | 4 | 5 |
| A2 | 2 | 3 | 3 |
| A3 | 3 | 1 | 2 |
How should the cargo consignments be assigned to the aeroplanes to maximize the profit?
A fair coin is tossed 12 times. Find the probability of getting exactly 7 heads .
Write the negation of the following statements :
(a) Chetan has black hair and blue eyes.
(b) ∃ x ∈ R such that x2 + 3 > 0.
Solve the following :
Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.
20 white rats are available for an experiment. Twelve rats are male. Scientist randomly selects 5 rats number of female rats selected on a specific day
Solve the following:
Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.
A highway safety group is interested in studying the speed (km/hrs) of a car at a check point.
Determine whether each of the following is a probability distribution. Give reasons for your answer.
| y | –1 | 0 | 1 |
| P(y) | 0.6 | 0.1 | 0.2 |
A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. If X denotes the age of a randomly selected student, find the probability distribution of X. Find the mean and variance of X.
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at least 3 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 (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.
10 balls are marked with digits 0 to 9. If four balls are selected with replacement. What is the probability that none is marked 0?
Defects on plywood sheet occur at random with the average of one defect per 50 sq.ft. Find the probability that such a sheet has:
- no defect
- at least one defect
Use e−1 = 0.3678
Solve the following problem :
It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on exactly 3 days of a week.
For the random variable X, if V(X) = 4, E(X) = 3, then E(x2) is ______
Let X be a discrete random variable. The probability distribution of X is given below:
| X | 30 | 10 | – 10 |
| P(X) | `1/5` | `3/10` | `1/2` |
Then E(X) is equal to ______.
Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.
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 Standard deviation of X.
The probability distribution of a random variable x is given as under:
P(X = x) = `{{:("k"x^2, "for" x = 1"," 2"," 3),(2"k"x, "for" x = 4"," 5"," 6),(0, "otherwise"):}`
where k is a constant. Calculate E(3X2)
