#### 12th Standard Maths English Medium Probability Distributions Reduced Syllabus Important Questions With Answer Key 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

Multiple Choice Questions

15 x 1 = 15
1. A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

(a)

6

(b)

4

(c)

3

(d)

2

2. On a multiple-choice exam with 3 possible destructives for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is

(a)

$\cfrac { 11 }{ 243 }$

(b)

$\cfrac { 3 }{ 8 }$

(c)

$\cfrac { 1 }{ 243 }$

(d)

$\cfrac { 5 }{ 243 }$

3. If P(X = 0) = 1-P(X = I). If E(X) = 3Var(X), then P(X = 0) is

(a)

$\cfrac { 2 }{ 3 }$

(b)

$\cfrac { 2 }{ 5 }$

(c)

$\cfrac { 1 }{ 5 }$

(d)

$\cfrac { 1 }{ 3 }$

4. If X is a binomial randam variable with expected value 6 and variance 2.4, then P(X=5) is

(a)

$\left( \cfrac { 10 }{ 5 } \right) \left( \cfrac { 3 }{ 5 } \right) ^{ 6 }\left( \cfrac { 2 }{ 5 } \right) ^{ 4 }$

(b)

$\left( \cfrac { 10 }{ 5 } \right) \left( \cfrac { 3 }{ 5 } \right) ^{ 10 }$

(c)

$\left( \cfrac { 10 }{ 5 } \right) { \left( \cfrac { 3 }{ 5 } \right) }^{ 4 }\left( \cfrac { 2 }{ 5 } \right) ^{ 6 }$

(d)

$\left( \cfrac { 10 }{ 5 } \right) \left( \cfrac { 3 }{ 5 } \right) ^{ 5 }\left( \cfrac { 2 }{ 5 } \right) ^{ 5 }$

5. The random variable X has the probability density function $f(x)=\left\{\begin{array}{ll} a x+b & 0<x<1 \\ 0 & \text { otherwise } \end{array}\right.$ and $E(X)=\cfrac { 7 }{ 12 }$ then a and b are respectively.

(a)

1 and $\cfrac { 1 }{ 2 }$

(b)

$\cfrac { 1 }{ 2 }$ and 1

(c)

2 and 1

(d)

1 and 2

6. Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, P(X = i) = k P(X = i-I) i = 1, 2 and P(X = 0) =$\cfrac { 1 }{ 7 }$ then the value of k is

(a)

1

(b)

2

(c)

3

(d)

4

7. Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is

(a)

0.24

(b)

0.48

(c)

0.6

(d)

0.96

8. If in 6 trials, X is a binomial variable which follows the relation 9P(X = 4) = P(X = 2), then the probability of success is

(a)

0.125

(b)

0.25

(c)

0.375

(d)

0.75

9. If $f(x)=\cfrac { 1 }{ 2 }$ ,$E\left( { x }^{ 2 } \right) =\cfrac { 1 }{ 4 }$ then var(x) is

(a)

0

(b)

$\cfrac { 1 }{ 4 }$

(c)

$\cfrac { 1 }{ 2 }$

(d)

1

10. In a binomial distribution, if the mean is 8 and the variance is 6, then the number of trials is

(a)

32

(b)

48

(c)

16

(d)

12

11. Adie is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of number of success is

(a)

$\cfrac { 8 }{ 3 }$

(b)

$\cfrac { 3 }{ 8 }$

(c)

$\cfrac { 4 }{ 5 }$

(d)

$\cfrac { 5 }{ 4 }$

12. Var (2x ± 5) is =________

(a)

5

(b)

var (2x) ± 5

(c)

4 var (X)

(d)

0

13. If the mean and variance of a binomial variate are 2 and 1 respectively, the probability that X takes a value greater than one is equal to__________.

(a)

$\cfrac { 5 }{ 16 }$

(b)

$\cfrac { 11 }{ 16 }$

(c)

$\cfrac { 10 }{ 16 }$

(d)

$\cfrac { 1 }{ 2 }$

14. A die is thrown 10 times. Getting a number greater than 3 is considered a success. The S.D of the number of successes is _________

(a)

2.5

(b)

1.56

(c)

5

(d)

25

15. For a Bernouli distribution

(a)

$\sigma =\sqrt { npq }$

(b)

$mean=\mu$

(c)

$\mu =p$

(d)

${ \sigma }^{ 2 }=pq$

1. 2 Marks

10 x 2 = 20
16. Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred

17. If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights
(i) exactly 10 will have a useful life of at least 600 hours;
(ii) at least 11 will have a useful life of at I least 600 hours;
(iii) at least 2 will not have a useful life of at : least 600 hours.

18. Compute P(X = k)for the binomial distribution, B(n,p) where
n=9,$p=\cfrac { 1 }{ 2 }$ ,k=7

19. Find the binomial distribution function for each of the following.
(i) Five fair coins are tossed once and X denotes the number of heads.
(ii) A fair die is rolled 10 times and X denotes the number of times 4 appeared.

20. Define a random variable

21. Define Probability Mass Function.

22. Define Probability Density function

23. A coin is tossed twice. If X is a random variable defined as the number of heads minus the number of tails, then obtain its probability distribution.

24. The probability distribution of a random variable X is given under :

Find (i) k
(ii) E(X)

25. Prove that Var(x+b)=Var(X)

1. 3 Marks

10 x 3 = 30
26. In a pack of 52 playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find the values of the random variable and number of points in its inverse images.

27. A six sided die is marked '2' on one face, '3' on two ofits faces, and '4' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the values of the random variable and number of points in its inverse images.

28. For the random variable X with the given probability mass function as below, find the mean and variance $\begin{matrix} f\left( x \right) \end{matrix}=\begin{cases} 2\left( x-1 \right) \qquad 1<x\ll 2 \\ 0\qquad otherwise \end{cases}$

29. The probability that a certain kind of component will survive a electrical test is $\cfrac { 3 }{ 4 }$ .
Find the probability that exactly 3 of the 5 components tested survive.

30. Suppose two coins are tossed once. If X denotes the number of tails,
(i) write down the sample space
(ii) find the inverse image of 1
(iii) the values of the random variable and number of elements in its inverse images

31. Two fair coins are tossed simultaneously (equivalent to a fair coin is tossed twice). Find the probability mass function for number of heads occurred.

32. If the probability mass function f (x) of a random variable X isx

 x 1 2 3 4 f (x) $\cfrac { 1 }{ 12 }$ $\cfrac { 5 }{ 12 }$ $\cfrac { 5 }{ 12 }$ $\cfrac { 1 }{ 12 }$

find (i) its cumulative distribution function, hence find
(ii) P(X ≤ 3) and,
(iii) P(X ≥ 2)

33. If X is the random variable with distribution function F(x) given by,
$F(x)=\begin{cases} \begin{matrix} 0 & x<0 \end{matrix} \\ \begin{matrix} x & 0\le x<1 \end{matrix} \\ \begin{matrix} 1 & 1\le x \end{matrix} \end{cases}$
then find (i) the probability density function f (x) (ii) P(0.2≤ X ≤0.7)

34. Find the mean and variance of a random variable X , whose probability density function is $f(x)=\begin{cases} \begin{matrix} { \lambda e }^{ -2x } & for\ge 0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}$

35. Consider a random variable X with p.d.$f(x)=\begin{cases} { 3x }^{ 2 },0<x<1 \\ 0,otherwise \end{cases}$Find Var(3X – 2).

1. 5 Marks

7 x 5 = 35
36. The probability density function of X is given
$f(x)=\begin{cases} \begin{matrix} { Ke }^{ \frac { -x }{ 3 } } & \begin{matrix} for & x>0 \end{matrix} \end{matrix} \\ \begin{matrix} 0 & \begin{matrix} for & x\le 0 \end{matrix} \end{matrix} \end{cases}$
Find
(i) the value of k
(ii) the distribution function.
(iii) P(X <3)
(iv) P(5 ≤X)
(v) P(X ≤ 4)

37. If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights
(i) exactly 10 will have a useful life of at least 600 hours;
(ii) at least 11 will have a useful life of at I least 600 hours; I
(iii) at least 2 will not have a useful life of at : least 600 hours.

38. A random variable X has the following probability mass function

 x 1 2 3 4 5 6 f(x) k 2k 6k 5k 6k 10k

Find
(i) P(2 < X < 6)
(ii) P(2 ≤ X < 5)
(iii) P(X ≤4)
(iv) P(3 < X )

39. The probability density function of random variable X is given by $f(x)=\begin{cases} \begin{matrix} k & 1\le x\le 5 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}$ Find
(i) Distribution function
(ii) P(X < 3)
(iii) P(2 < X < 4)
(iv) P(3 ≤ X )

40. A box contains 4 red and 5 black marbles. Find the probability distribution of the red marbles in a random draw of three marbles. Also find the mean and standard deviation of the distribution

41. If the sum and the product of the mean and variance of a binomial distribution are 1.8 and 0.8 respectively, find the probability distribution and the probability of at least one success.

42. Ten coins are tossed simultaneously. What is the probability of getting (a) exactly 6 heads (b) at least 6 heads (c) at most 6 heads?