What is student As expected value for this game Two students

Two students, A and B, play a game of chance: student A pays $2 to play, lie is then allowed to flip a coin at most 5 times. If on the first time tails appears, student A get a $1 and the game ends, Otherwise, student A flips again, if tails appears, he gets $2. They continue so that if the first time tails appears is on the third toss student A gets $4, the fourth toss, $8, and the fifth toss $16. What is student A\'s expected value for this game?

Solution

Expected Value for a Discrete Random Variable

E(X)=xipi

probability of getting head 1st = 1/2

probability of getting tail at 1st fip = 1/2

as the number of flip increases the probability will become its power

like the probability of tail in 2nd flip = 1/2^2=1/4

same for head and also for other number of flips

now the expected value will be

E(X) = 1/2*1 + 1/2*2 +1/2^2*4 + 1/2^3*8 + 1/2^4*16 = 4.5