A coin toss has possible outcomes H and T with probabilities 3/4 and 1/4 respectively. A gambler observes a sequence of tosses of this coin until H occurs. Let the first H occur on the nth toss. If n is odd, then the gambler’s prize is −2^n , and if n is even, then the gambler’s prize is 2^n . What is the expected value of the gambler’s prize?
Answers
Given : A coin toss has possible outcomes H and T with probabilities 3/4 and 1/4 respectively.
A gambler observes a sequence of tosses of this coin until H occurs.
Let the first H occur on the nth toss.
If n is odd, then the gambler’s prize is −2^n , and if n is even, then the gambler’s prize is 2^n .
To Find : What is the expected value of the gambler’s prize
Solution:
H occurs on nth toss
Expected value in case of odd
= (-2)¹ . (3/4)¹ + (-2)³.(3/4)(1/4)² + _____
= -2 * (3/4) [ 1 + 2²/4² + 2⁴/4⁴ + ______ ]
= ( -3/2 )[ 1 + 2²/4² + 2⁴/4⁴ + ______ ]
Expected value in case of even
= ( 2)²(3/4)(1/4) + ( 2)⁴(3/4)(1/4)³ + ____
= 2² *(3/4)(1/4) [ 1 + 2²/4² + 2⁴/4⁴ + ______ ]
= (3/4) [ 1 + 2²/4² + 2⁴/4⁴ + ______ ]
Net expected value of the gambler’s prize
= ( -3/2 )[ 1 + 2²/4² + 2⁴/4⁴ + ______ ] + (3/4) [ 1 + 2²/4² + 2⁴/4⁴ + _____]
= (-3/4) [ 1 + 2²/4² + 2⁴/4⁴ + ______ ]
= (-3/4) [ 1 + 1/2² + 1/2⁴ + ______ ]
a = 1 r = 1/4
(-3/4) ( 1/( 1 - 1/4))
= (-3/4) ( 1/(3/4))
= - 1
expected value of the gambler’s prize = - 1
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