Question

A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, then she wins one-half of the value that appears on the die. Determine her expected winnings.

Answer 1

Answer:

4.375

Explanation:

Let random variable be X represent the result of toss coin

Let Y be the random variable represent number on dice.

Probability of getting heads = P(H) = 1/2

Probability of getting tails = P(T) = 1/2

Probability of getting number ob die

P(1) = P(2) = P(3) = P(4) = P(5)  =P(6) = 1/6

Expected winning when coin landed head

$E_{Head}(\dfrac{Y}{X}) = E(2Y).P(X = H)$

=$2. \dfrac{1}{2}(1.\dfrac{1}{6}+2.\dfrac{1}{6}+3.\dfrac{1}{6}.......+6..\dfrac{1}{6})$

= 3.5

Expected winning when coin landed tails

$E_{Tails}(\dfrac{Y}{X}) = E(\dfrac{1}{2}Y).P(X = T)$

=$\dfrac{1}{2}. \dfrac{1}{2}(1.\dfrac{1}{6}+2.\dfrac{1}{6}+3.\dfrac{1}{6}.......+6..\dfrac{1}{6})$

= 0.875

Expected winning

= $E_{Head}(\dfrac{Y}{X}) + E_{Tails}(\dfrac{Y}{X})$

= 3.5 + 0.875

= 4.375