# MCA DCA7101 PROBABILITY AND STATISTICS

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SET 1

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(A|B) = P(B|A) * P(A) / P(B), where P(A|B) is the probability of event A given event B is true, P(B|A) 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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1. 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+ a2+2a2+7a2

=5a+10a2

=5(1/10)+10(1/10)

1. The daily earning of a vendor for a period of 43 days are given below

 Daily earning (Rs.) 118-126 127-135 136-144 145-153 154-162 163-171 172-180 No. of days 3 8 9 12 5 4 2

Calculate Standard Deviation and coefficient of variation.

Solution:

 Daily earnings f x fx (x-x̄) F(x- x̄) f(x- x̄)2 118-126 3 122 366 -23.86 -71.58 1707.90 127-135 8 131 1048 -14.86 -118.88 1766.56

SET-II

1. 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-

1. Exactly two will strike the target.
2. 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) * ((1-p)^(n-k))

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 (1-p

1. 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,

1. 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.

1. Laspeyre’s Index
2. 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)