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

Answer:

E (Y) = 3

Step-by-step 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) = $\frac{1}{2}$

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) = $\frac{1}{2}$

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 = $\frac{1}{4}$

Y|U ≅ geometric = $\frac{1}{2}$

also; x ≅ geometric (p)

∴ E (x) = $\frac{1}{p}$

⇒ $\frac{Y}{T}$ = 4 ; also $\frac{Y}{U}$ = 2

E (Y) = 4 × $\frac{1}{2}$ + 2 ×

        = 2+1

E (Y) = 3