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Option 2

#### For three events A, B and C, P(Exactly one of A or B occurs) =P(Exactly one of B or C occurs) =P(Exactly one of C or A occurs)     and P(All the three events occur simultaneously)  Then the probability that at least one of the events occurs, is :   Option: 1 $\frac{7}{16}$ Option: 2 $\frac{7}{64}$ Option: 3 $\frac{3}{16}$ Option: 4 $\frac{7}{32}$

P (exactly one of A or B) =P(exactly B or C)

=P (exactly one of A or C) =

P(A) + P (B) - 2P (AB)=

P(B) +P (C) - 2P (BC)=

P(C)+P(A)-2P (CA)=

P (A) + P (B) -2 P (AB) - P(BC)-P(CA) =

P(ABC)=

P(ABC)=

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#### Let A and B be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then, which of the following is TRUE? Option: 1 Option: 2 Option: 3 Option: 4

A & B are independent events

So, A & B' are also independent events, and hence

$P\left(\frac{A}{B^{\prime}}\right)=P(A)=\frac{1}{3}$

Correct Option (2)

#### The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by q, where and If the new mean and new s.d. become half of their original values, then q is equal to :Option: 1Option: 2Option: 3Option: 4

Some Important Point Regarding Statistics -

Some Important Point Regarding Statistics

1. The sum of the deviation of an observation from their mean is equal to zero. i.e. .
2. The sum of the square of the deviation from the mean is minimum, i.e.
3. The mean is affected accordingly if the observations are given a mathematical treatment i.e. addition, subtraction multiplication by a constant term.
4. If set of n1 observations has mean  and set of n2 observations has mean , then their combined mean is  and the .
5. If set of  n1 observations has mean  and set of n2 observations has mean  then their combined variance is given by

where,

are the means and   are the standard deviations of two series.

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Old Mean

Old Standard Deviation

If each observation is multiplied with p & then q is subtracted

New mean is the half of old mean

and new standard deviations is the half of old mean

Case 1

Using equation 1

we get

Case 1

Using equation 1

we get

Correct Option (1)

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#### If the variance of the first n natural numbers is 10 and the variance of the first m even natural numbers is 16, then m+n is equal to _____. Option: 1 12 Option: 2 18 Option: 3 16 Option: 4 20

Dispersion (Variance and Standard Deviation) -

Variance and Standard Deviation

The mean of the squares of the deviations from the mean is called the variance and is denoted by σ2 (read as sigma square).

Variance is a quantity which leads to a proper measure of dispersion.

The variance of n observations x1 , x2 ,..., xn is given by

Standard Deviation

The standard deviation is a number that measures how far data values are from their mean.

The positive square-root of the variance is called standard deviation. The standard deviation, usually denoted by σ  and it is given by

-

variance of the first n natural numbers is 10

n = 1,2,3,4........n

Using variance formula

variance of the first m even natural numbers is 16

N = 2,4,6,8........2m

N = 2(1,2,3,4........m)

Using variance formula

Hence

#### An unbiased coin is tossed 5 times.Suppose that a variable X is assigned the value k when consecutive heads are obtained for k = 3,4,5, otherwise X takes the value -1. Then the expected value of X, is : Option: 1 Option: 2 Option: 3 Option: 4

An unbiased coin is tossed 5 times, so total number of outcome is 25 = 32.

Now, the probability for getting number of heads occurring simultaneously

$\\\mathrm{for\;k=0}\\\mathrm{E(k)=\{T T T T T\}=1}\\\mathrm{P(k)=\frac{1}{32}}$

$\\\mathrm{for\;k=1}\\\mathrm{E(k)=\{ HTTTT,THTTT,TTHTT,TTTHT,TTTTH,HTHTT,HTTHT,}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;HTTTH,THTHT,THTTH,TTHTH,HTHTH\} =12}\\\mathrm{P(k)=\frac{12}{32}}$

$\\\mathrm{for\;k=2}\\\mathrm{E(k)=\{ { HHTTT,THHTT,TTHHT,TTTHH,HHTHT,HHTTH },}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; { THHTH,HTHHT,HTTHH,THTHH,HHTHH }\} =11}\\\mathrm{P(k)=\frac{11}{32}}$

$\\\mathrm{for\;k=3}\\\mathrm{E(k)=\{ HHHTT,THHHT,TTHHH,HHHTH,HTHHH\} =5}\\\mathrm{P(k)=\frac{5}{32}}$

$\\\mathrm{for\;k=4}\\\mathrm{E(k)=\{ \mathrm{HHHHT,THHHH}\} =2}\\\mathrm{P(k)=\frac{2}{32}}$

$\\\mathrm{for\;k=5}\\\mathrm{E(k)=\{ \mathrm{HHHHH}\} =1}\\\mathrm{P(k)=\frac{1}{32}}$

$\begin{array}{|c|c|c|c|c|c|c|} \hline \mathrm{K} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \mathrm{P}(\mathrm{k}) & \frac{1}{32} & \frac{12}{32} & \frac{11}{32} & \frac{5}{32} & \frac{2}{32} & \frac{1}{32} \\ \hline \end{array}$

$\text { For } \mathrm{k}=0,1,2 ;\mathrm{X}=-1 \text { and for } \mathrm{k}=3,4,5; \mathrm{X}=\mathrm{k} \text { . }$

Now expected value is

$\\\sum X \times P(k)=(-1) \times \frac{1}{32}+(-1) \times \frac{12}{32}+(-1) \times \frac{11}{32} +3 \times \frac{5}{32}+4 \times \frac{2}{32}+5 \times \frac{1}{32}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=\frac{1}{8}$

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#### The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is :Option: 1Option: 2Option: 3Option: 4

Dispersion (Variance and Standard Deviation) -

Variance and Standard Deviation

The mean of the squares of the deviations from the mean is called the variance and is denoted by σ2 (read as sigma square).

Variance is a quantity which leads to a proper measure of dispersion.

The variance of n observations x1 , x2 ,..., xn is given by

Standard Deviation

The standard deviation is a number that measures how far data values are from their mean.

The positive square-root of the variance is called standard deviation. The standard deviation, usually denoted by σ  and it is given by

Variance and Standard Deviation of a Discrete Frequency Distribution

The given discrete frequency distribution be

Variance and Standard deviation of a continuous frequency distribution

The formula for variance and standard deviation are the same as in the case of discrete frequency distribution. Here, is the mid point of each class.

Another formula for Standard Deviation

Shortcut method to find variance and standard deviation

The values of   in a discrete distribution or the mid points  of different classes in a continuous distribution are large and so the calculation of mean and variance becomes tedious and time consuming.

Here is the shortcut method to find variance and standard deviatio

Let the assumed mean be ‘A’ and the scale be reduced to 1/h times (h being the width of class-intervals).

Let the step-deviations or the new values be .

Replacing x_i from (1) in (2),

Now Variance of the variable x

From (3) and (4),

-

Correct Option (2)

#### Let A and B be two events such that the probability that exactly one of them occurs is  and the probability that A or B occurs is  then the probability pf both of them occur together is :  Option: 1Option: 2Option: 3Option: 4

Correct Option (1)

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#### If the mean and variance of eight numbers 3,7,9,12,13,20,x and y be 10 and 25 respectively, then  is equal to _____.Option: 1 54Option: 2 60Option: 3 24Option: 4 None

Mean -

Mean of the Ungrouped Data

If n observations in data are x1, x2, x3, ……, xn, then arithmetic mean  is given by

-

Dispersion (Variance and Standard Deviation) -

Variance and Standard Deviation

The mean of the squares of the deviations from the mean is called the variance and is denoted by σ2 (read as sigma square).

Variance is a quantity which leads to a proper measure of dispersion.

The variance of n observations x1 , x2 ,..., xn is given by

-

#### In a workshop, there are five machines and the probability of any one of them to be out of service on a day is . If the probability that at most two machines will be out of service on the same day is  then k is equal to :Option: 1Option: 2Option: 3Option: 4

Required probability = when no. the machine has fault + when only one machine has fault + when only two machines have the fault.

Correct Option (4)