subject: Expected Value Equation [print this page] Expected Value Equation: Expected Value Equation:
If in an observation there are n possible ways exhaustive and mutually exclusive and out of them in m ways in the event A occurs, then the probability of occurrence of event A is given by P(A) =m / n.If a variable variable x can take any of values (x1, x2, ..xn) with the corresponding probabilities (P1, P2,....Pn) then expected value equation of x or exception of x is written as, E(x) = P1X1+ P2X2+......+PnXn.
Example Problems on Expected Value Equation:
Expected value equation - problem 1:
The probabilities of a specific problem being solved independently by A and B are and 1/3 respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved
(ii) exactly one of them solves the problem.
Solution:
Let E1 = event that A solves the problem,
And E2 =event that B solves the problem.
Then, P(E1) = 1/2 and P(E2) = 1/3
P('barE' 1) = (1 1/2 ) = 1/2
P('barE' 2) = (1 1/3) = 2/3.
Clearly, E1 and E2 are independent events.
Therefore P(E1 'nn' E2) = P(E1) x P(E2) = (1/2 x 1/3) = 1/6.
(i) P(the problem is solved)
= P(at least one of A and B solves the problem)
= P(E1 or E2) = P(E1 'uu' E2)
= P(E1) + P(E2) P(E1'nn' E2)
= (1/2 +1/3 1/6)
= 4/6
= 2/3.
(iii)P(exactly one of them solves the problem)
=P[(E1 and not E2) or (E2 and not E1)]
=P(E1 and not E2) + P(E2 and not E1)
=P(E1'nn' 'barE' 2) + P(E2'nn' 'barE' 1)
=P(E1) x P('barE' 2) + P(E2) x P('barE' 1)
= (1/2 x 2/3) + (1/3 x 1/2)
= (1/3 + 1/6)
= 3/6
= 1/2
Expected value equation - problem 2:
A and B appear for an interview for two posts. The probability of As selection is (1/3) and that of Bs selection is (2/5). Find the probability that only one of them will be selected.
Solution:
Let E1 = event that A is selected,
And E2 = event that B is selected.
Then, P(E1) = 1/3 and P(E2) = 2/5.
P('barE' 1) = (1-1/3)
= 2/3
P('barE' 2) = (1-2/5)
= 3/5.
Therefore P(event that only one of them is selected)
= P[(E1 and not E2) or (E2 and not E1)]
=P[(E1'nn' 'barE' 2) or (E2'nn' 'barE' 1)]
=P(E1'nn' 'barE' 2) + P(E2'nn' 'barE' 1)
=P(E1) * P('barE' 2) + P(E2) * P('barE' 1)
=(1/3 x 3/5) + (2/5 x 2/3)
= (1/5 + 4/15)
= 7/15.
Practice Problem on Expected Value Equation:
1. The odds against a man who is 45 years old, living till he is 70 are 7 : 5, and the odds against his wife who is now 36, living till she is 61 are 5 : 3. Find the probability that
(i) The couple will be alive 25 years hence
(ii) At least one of them will be alive 25 years hence.
