Advertisements
Advertisements
प्रश्न
Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
उत्तर
Let X denote the number of aces in a sample of 2 cards drawn from a well-shuffled pack of 52 playing cards. Then, X can take the values 0, 1 and 2.
Now,
\[P\left( X = 0 \right)\]
\[ = P\left( \text{ no ace }\right)\]
\[ = \frac{{}^{48} C_2}{{}^{52} C_2}\]
\[ = \frac{2256}{2652}\]
\[ = \frac{188}{221}\]
\[P\left( X = 1 \right)\]
\[ = P\left( 1 \text{ ace } \right)\]
\[ = \frac{{}^4 C_1 \times^{48} C_1}{{}^{52} C_2}\]
\[ = \frac{192}{1326}\]
\[ = \frac{32}{221}\]
\[P\left( X = 2 \right)\]
\[ = P\left( 2 \text{ aces} \right)\]
\[ = \frac{{}^4 C_2}{{}^{52} C_2}\]
\[ = \frac{6}{1326}\]
\[ = \frac{1}{221}\]
Thus, the probability distribution of X is given by
| X | P (X) |
| 0 |
\[\frac{188}{221}\]
|
| 1 |
\[\frac{32}{221}\]
|
| 2 |
\[\frac{1}{221}\]
|
APPEARS IN
संबंधित प्रश्न
A random variable X has the following probability distribution:
then E(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 |
State the following are not the probability distributions of a random variable. Give reasons for your answer.
| Z | 3 | 2 | 1 | 0 | -1 |
| P(Z) | 0.3 | 0.2 | 0.4 | 0.1 | 0.05 |
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).
The probability distribution function of a random variable X is given by
| xi : | 0 | 1 | 2 |
| pi : | 3c3 | 4c − 10c2 | 5c-1 |
where c > 0 Find: c
A random variable X takes the values 0, 1, 2 and 3 such that:
P (X = 0) = P (X > 0) = P (X < 0); P (X = −3) = P (X = −2) = P (X = −1); P (X = 1) = P (X = 2) = P (X = 3) . Obtain the probability distribution of X.
Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement, from a bag containing 4 white and 6 red balls
From a lot containing 25 items, 5 of which are defective, 4 are chosen at random. Let X be the number of defectives found. Obtain the probability distribution of X if the items are chosen without replacement .
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.
The probability distribution of a random variable X is given below:
| x | 0 | 1 | 2 | 3 |
| P(X) | k |
\[\frac{k}{2}\]
|
\[\frac{k}{4}\]
|
\[\frac{k}{8}\]
|
Find P(X ≤ 2) + P(X > 2) .
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 |
A pair of fair dice is thrown. Let X be the random variable which denotes the minimum of the two numbers which appear. Find the probability distribution, mean and variance of X.
A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.
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.
For what value of k the following distribution is a probability distribution?
| X = xi : | 0 | 1 | 2 | 3 |
| P (X = xi) : | 2k4 | 3k2 − 5k3 | 2k − 3k2 | 3k − 1 |
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:
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 .
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
From a lot of 15 bulbs which include 5 defective, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence, find the mean of the distribution.
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 exactly 7 heads .
Determine whether each of the following is a probability distribution. Give reasons for your answer.
| x | 0 | 1 | 2 |
| P(x) | 0.4 | 0.4 | 0.2 |
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 X > 1
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 :
The probability that a bomb will hit the target is 0.8. Find the probability that, out of 5 bombs, exactly 2 will miss the target.
Solve the following problem :
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. Find the probability that the inspector finds at most one defective item in the 4 selected items.
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.
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 |
Variance of X.
The random variable X can take only the values 0, 1, 2. Given that P(X = 0) = P(X = 1) = p and that E(X2) = E[X], find the value of p
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 E(X)
A bag contains 1 red and 3 white balls. Find the probability distribution of the number of red balls if 2 balls are drawn at random from the bag one-by-one without replacement.
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`
