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SET 1
 Suppose a person throws a die. If he gets a 5 or 6, he tosses a coin three times and notes the number of heads. If he gets 1, 2, 3 or 4, tosses a coin once and notes whether a head or tail is obtained. If he obtained exactly one head; what is the probability that he threw 5 or 6 with the die.
Ans 1.
The solution to this problem involves Bayes’ theorem, which states that P(AB) = P(BA) * P(A) / P(B), where P(AB) is the probability of event A given event B is true, P(BA) is the probability of event B given event A is true, P(A) is the probability of event A, and P(B) is the probability of event B.
We want to find the probability that the die
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 A random variable has the following probability distribution
Find (i) the value of ‘’
(iii)
(i)
Solution: Since it is a probability distribution, so, sum of all probabilities must be 1
Since, “a” is a probability, so, it cannot be negative
A = 0.1
(ii) [1.5<x<4.5/x>2]
When 1.5<x<4.5 =2a+3a
=5a
=5(1/10)
=1/2 or 0.5
When x>2 =P(X=3)+ P(X=4)+ P(X=5)+ P(X=6)+ P(X=7)
= 2a+3a+ a^{2}+2a^{2}+7a^{2}
=5a+10a^{2}
=5(1/10)+10(1/10)
 The daily earning of a vendor for a period of 43 days are given below
Daily earning (Rs.)  118126  127135  136144  145153  154162  163171  172180 
No. of days  3  8  9  12  5  4  2 
Calculate Standard Deviation and coefficient of variation.
Solution:
Daily earnings  f  x  fx  (xx̄)  F(x x̄)  f(x x̄)^{2} 
118126  3  122  366  23.86  71.58  1707.90 
127135  8  131  1048  14.86  118.88  1766.56 
SETII
 The probability that a bomb dropped from a plane will strike the target is 1/5. If 6 bombs are dropped, find the probability that
 Exactly two will strike the target.
 Atleast two will strike the target.
Ans 4.
To answer this question, we’ll need to use the binomial probability formula, which is as follows:
P(X=k) = C(n, k) * (p^k) * ((1p)^(nk))
where: P(X=k) is the probability of k successes in n trials, C(n, k) is the number of combinations of n items taken k at a time (also known as “n choose k”), p is the probability of success on a given trial, and (1p
 Compute the regression equation of Y on X and regression equation of X on Y on the basis of the following information
X  Y  
Mean  40  45 
Standard Deviation  10  9 
The correlation coefficient between X and Y is 0.50.
Also estimate the value of Y for X = 48, using the appropriate regression equation.
Ans 5.
Given that,
 The following data relate to the prices and quantities of 4 commodities in the years 1982 and 1983. Construct the following index numbers of price for the year 1983 by using 1982 as base year.
 Laspeyre’s Index
 Paasche’s Index
iii. Fisher’s Index
Commodity  1982  1983  
Price  Quantity  Price  Quantity  
A  5  100  6  150 
B  4  80  5  100 
C  2.5  60  5  72 
D  12  30  9  33 
Ans 6.
Solution:
Commodit Y

1982  1983

P1q0  P0q0  P1q1  P0q1  
Price(P0)  Quantity(Q0)  Price(P1)  Quantity(Q1)  