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Suppose in a game of coin tossing a person, say X will get Rs. 5 if the head turns up and will lose Rs. 4 if tail turns up. If n (number of trails) in which an unbiased coin is tossed are considered, what is the value of his expectation?
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Solution | |
"x" indicate Mr. X's gain [Since you are required to find the expected gain of X, the variable would represent Mr. X's gain.] In a single tossing of the coin, the amount won by Mr. X may be
⇒ The values carried by the variable ("x") would be either − 4 or + 5
"X" represents the random variable and P(X = x) represents the probability that the value within the range of the random variable is a specified value of "x" In the experiment of tossing a coin
Therefore, probability that the amount won by Mr. X would be
The probabilty distribution of "x" would be
Calculations for Mean and Standard Deviations
Expected winnings of Mr. X
Let x1, x2, x3, ... represent the amount won by Mr. X on the first throw, second throw, third throw, .... Mr. X's expected amount of winning
Each trial (throwing of the coin) is identical and therefore the expected amount of winning in each trial would be the same Therefore, Mr. X's expected winning in "n" trials (throwings) of the coin = E (x1) + E (x2) + E (x3) + ... + E (xn)
AlternativeWhere all the trials are identical the expected amount of winning in all the trials together is given by Number of Trials × Expectation in each trial ⇒ Expected amount of winnings in "n" tosses of the coin
= No. of times the coin is tossed × Expected winnings per toss
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Credit : Vijayalakshmi Desu |