Question

You roll a 4-sided die repeatedly. On your odd-numbered rolls (1st,3rd,5th, etc.) you are victorious if you get a 4. On your even-numbered rolls, you are victorious if you get a 3 or 4. You stop once you are victorious. Let Y be the number of times you roll.

Find E[Y].

Answer 1

E (Y) = 3

Explanation:

If a 4-sided die is being rolled repeatedly; and the odd-numbered rolls (1st 3rd,5th, etc.)

The probability of odd number roll will be, p(T) =
`(1)/(2)`

However, on your even-numbered rolls, you are victorious if you get a 3 or 4. Also, the probability of even number roll, p(U) =
`(1)/(2)`

In order to calculate: E (Y); We can say Y to be the number of times you roll.

We know that;

E (Y) = E ( Y|T ) p(T) + E ( Y|U ) p(U)

Let us calculate E ( Y|T ) and E ( Y|U )

Y|T ≅ geometric =
`(1)/(4)`

Y|U ≅ geometric =
`(1)/(2)`

also; x ≅ geometric (p)

∴ E (x) =
`(1)/(p)`

⇒
`(Y)/(T)` = 4 ; also
`(Y)/(U)` = 2

E (Y) = 4 ×
`(1)/(2)` + 2 ×

= 2+1

E (Y) = 3