Page 342 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 26: Theory of Probability: Introduction and Uses

Solution: The distribution of earnings (X) is given as: Notes

X X = 80,000 X = 95,000
1 2
P 1 – 0.76 = 0.24 0.76

E (X) = 80,000 × 0.24 + 95,000 × 0.76
= Rs. 91,400
Example 21: A player tosses 3 fair coins. He wins Rs. 12 if 3 heads appear, Rs. 8 if 2 heads appear
and Rs. 3 if 1 head appears. On the otherhand, he loses Rs. 25 if 3 tails appear. Find the
expected gain of the player.
Solution: If p denotes the probability of getting a head and X denotes the corresponding amount
of winning, then the distribution of X is given by:
Heads: 0H 1H 2H 3H

Favourable Events TTT HTT, THT, TTH HHT, HTH, THH HHH

1 1 1 1 1 1 1 3 1 1 1 3 1 1 1 1
P × × = + + = + + = × × =
2 2 2 8 8 8 8 8 8 8 8 8 2 2 2 8

X – 25 3 8 12
Winning amount

The expected gain of the player is given by:
E (X) = ( 1 )− +25 3 () + 3 3 () + 8 1 ( )12
8 8 8 8
− +25 9 + +24 12 20 5
= = = = Rs. 2.50.
8 8 2
Example 22: A player tosses two fair coins. He wins Rs. 5 if 2 heads appear, Rs. 2 if one head appear
and Rs. 1 if no head appear. Find his expected gain of the player.
Solution: If p denotes the probability of getting a head and X denotes the corresponding amount
of winning, then the probability distribution of X is given by:

Heads: 0H 1H 2H
Favourable Events TT HT, TH HH

1 1 1 1 1 1 1 1 1
P × = + = × =
2 2 4 4 4 2 2 2 4
X 1 2 5

The expected gain of the player is given by:
E (X) = P X + P X + P X
1 1 2 2 3 3
1 1 1
1
= ×+ × 2 + × 5 = Rs. 2.50
4 2 4

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