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 number greater than 4.
उत्तर
Success is defined as a number greater than 4 appears on at least one die.
Let X denote the number of successes.
∴ Possible values of X and 0, 1, 2.
Let, P(getting a number greater than 4) = p = `(2)/(6) = (1)/(3)`
∴ q = 1 – p = `1 - (1)/(3) = (2)/(3)`
∴ P(X = 0) = P(no success) = qq = q2 = `(4)/(9)`
9 P(X = 1) = P(one success) = qp + qp = 2pq
= `(2 xx 1)/(3) xx (2)/(3)`
= `(4)/(9)`
P(X = 2) = P(two successes) = pp = p2 = `(1)/(9)`
∴ Probability distribution of X is as follows:
| X | 0 | 1 | 2 |
| P(X = x) | `(4)/(9)` | `(4)/(9)` | `(1)/(9)` |
APPEARS IN
संबंधित प्रश्न
From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain -
(a) exactly 1 defective bulb.
(b) at least 1 defective bulb.
State the following are not the probability distributions of a random variable. Give reasons for your answer.
| X | 0 | 1 | 2 |
| P (X) | 0.4 | 0.4 | 0.2 |
State the following are not the probability distributions of a random variable. Give reasons for your answer.
| X | 0 | 1 | 2 | 3 | 4 |
| P(X) | 0.1 | 0.5 | 0.2 | -0.1 | 0.3 |
Find the probability distribution of number of tails in the simultaneous tosses of three coins.
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4
(ii) six appears on at least one die
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.
The random variable X has probability distribution P(X) of the following form, where k is some number:
`P(X = x) {(k, if x = 0),(2k, if x = 1),(3k, if x = 2),(0, "otherwise"):}`
- Determine the value of 'k'.
- Find P(X < 2), P(X ≥ 2), P(X ≤ 2).
An urn contains 25 balls of which 10 balls bear a mark ‘X’ and the remaining 15 bear a mark ‘Y’. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that
(i) all will bear ‘X’ mark.
(ii) not more than 2 will bear ‘Y’ mark.
(iii) at least one ball will bear ‘Y’ mark
(iv) the number of balls with ‘X’ mark and ‘Y’ mark will be equal.
A random variable X has the following probability distribution:
| Values of X : | −2 | −1 | 0 | 1 | 2 | 3 |
| P (X) : | 0.1 | k | 0.2 | 2k | 0.3 | k |
Find the value of k.
Two dice are thrown together and the number appearing on them noted. X denotes the sum of the two numbers. Assuming that all the 36 outcomes are equally likely, what is the probability distribution of X?
A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 17, 20, 17, 16, 19 and 20 years respectively. One student is selected in such a manner that each has the same chance of being selected and the age X of the selected student is recorded. What is the probability distribution of the random variable X?
Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.
Two cards are drawn simultaneously from a well-shuffled deck of 52 cards. Find the probability distribution of the number of successes, when getting a spade is considered a success.
From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability distribution of the number of defective bulbs.
Find the mean and standard deviation of each of the following probability distribution :
| xi : | -3 | -1 | 0 | 1 | 3 |
| pi : | 0.05 | 0.45 | 0.20 | 0.25 | 0.05 |
A discrete random variable X has the probability distribution given below:
| X: | 0.5 | 1 | 1.5 | 2 |
| P(X): | k | k2 | 2k2 | k |
Find the value of k.
A fair coin is tossed four times. Let X denote the number of heads occurring. Find the probability distribution, mean and variance of X.
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.
A random variable X takes the values 0, 1, 2, 3 and its mean is 1.3. If P (X = 3) = 2 P (X = 1) and P (X = 2) = 0.3, then P (X = 0) is
Mark the correct alternative in the following question:
The probability distribution of a discrete random variable X is given below:
| X: | 2 | 3 | 4 | 5 |
| P(X): |
\[\frac{5}{k}\]
|
\[\frac{7}{k}\]
|
\[\frac{9}{k}\]
|
\[\frac{11}{k}\] |
The value of k is .
For the following probability density function (p. d. f) of X, find P(X < 1) and P(|x| < 1)
`f(x) = x^2/18, -3 < x < 3`
= 0, otherwise
The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class :
(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )
A random variable X has the following probability distribution :
| X = x | -2 | -1 | 0 | 1 | 2 | 3 |
| P(x) | 0.1 | k | 0.2 | 2k | 0.3 | k |
Find the value of k and calculate mean.
A fair coin is tossed 12 times. Find the probability of getting at least 2 heads .
If random variable X has probability distribution function.
f(x) = `c/x`, 1 < x < 3, c > 0, find c, E(x) and Var(X)
Find the premium on a property worth ₹12,50,000 at 3% if the property is fully insured.
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 pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes
Solve the following problem :
Following is the probability distribution of a r.v.X.
| x | – 3 | – 2 | –1 | 0 | 1 | 2 | 3 |
| P(X = x) | 0.05 | 0.1 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |
Find the probability that X is non-negative
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.
Solve the following problem :
In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.
Calculate the probabilities of obtaining an answer yes from all of the selected students.
For the random variable X, if V(X) = 4, E(X) = 3, then E(x2) is ______
Consider the probability distribution of a random variable X:
| X | 0 | 1 | 2 | 3 | 4 |
| P(X) | 0.1 | 0.25 | 0.3 | 0.2 | 0.15 |
Calculate `"V"("X"/2)`
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 P(X ≥ 4)
For the following probability distribution:
| X | 1 | 2 | 3 | 4 |
| P(X) | `1/10` | `3/10` | `3/10` | `2/5` |
E(X2) is equal to ______.
A large chain retailer purchases an electric device from the manufacturer. The manufacturer indicates that the defective rate of the device is 10%. The inspector of the retailer randomly selects 4 items from a shipment. Complete the following activity to find the probability that the inspector finds at most one defective item in the 4 selected items.
Solution:
Here, n = 4
p = probability of defective device = 10% = `10/100 = square`
∴ q = 1 - p = 1 - 0.1 = `square`
X ∼ B(4, 0.1)
`P(X=x)=""^n"C"_x p^x q^(n-x)= ""^4"C"_x (0.1)^x (0.9)^(4 - x)`
P[At most one defective device] = P[X ≤ 1]
= P[X=0] + P[X=1]
= `square+square`
∴ P[X ≤ 1] = `square`
