23 A student rolls a die until the first \"4\" appears. Let X be the numbers of

Question

23 A student rolls a die until the first "4" appears. Let X be the numbers of roll's required until (and including) this first "4". After this is completed, he begins rolling again until he gets a "3". Let Y be the number of rolls, after the first "4", up to (and including) the next "3". For example, if the sequence of rolls is 213062341201013. then X = 8 and Y - 7. Find the expectations for X and Y. Chris tries to throw a ball of paper in the wastebasket behind his back (without looking). He estimates that his chance of success each time, regardless of the outcome of the other attempts, is 1/3. Let X be the number of attempts required. If he is not successful within the first. 5 attempts, then he quits, and ho lets X - (i in such a case. Find E[X].

Explanation / Answer

2. P(4) = 1/6

First 4 can appear in the 1st roll, 2nd roll, 3rd roll, .......

So,

E(X) = 1/6 + 2* 1/6 * 5/6 + 3* 1/6 * (5/6)^2 + ...... Equ (1)

E(X) * 5/6 = 1/6 * 5/6 + 2* 1/6 * (5/6)^2 + ...... Equ (2) = Equ (1) * 5/6

Equ (1) - Equ (2):

E(X) * 1/6= 1/6 + 1/6* 5/6 + 1/6* (5/6)^2 + ......

E(X) = 6* [(1/6)/(1 - 5/6)]

E(X) = 6

3. P(Success) = 1/3

So,

E(X) = 1/3 + 2* 2/3 * 1/3 + 3* (2/3)^2 * 1/3 + 4* (2/3)^3 * 1/3 + 5* (2/3)^4 * 1/3 + 6* (2/3)^6

E(X) = 2.47

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