The “z-number” of a number x is defined as the ones digit of 2^x. Antony rolls a die with 6 sides labeled with the integers from 1 to 6, each of which has an equal probability of landing face-up. He then takes 3^z, where z is the z-number of the result of his die roll, and plots 3^z on a number line as the point A. Finally, he repeats this entire process, this time plotting the result as the point B. What is the probability that the distance between A and B is greater than the value of B? Choose the correct choice: A) 3/8 B) 13/36 C) 17/36 D) 19/36 E) 23/36
If you calculate 3^z for 1st roll, all 6 results will be 9, 81, 6561, 729, 9, 81. This result is the same for 2nd roll. 9, 81, 6561, 729, 9, 81.   About distance: If the first result is 9 and the second is also 9, the distance is   9-9= 
<gadget id="calculator">  9-9</gadget>
<output>0</output>
0 which is smaller than 9. If the first result is 9 and the second is 81, the distance is   81-9= 
<gadget id="calculator">  81-9</gadget>
<output>72</output>
72 which is also smaller than B which has the value of 81. If the first result is 81 and the second is 9, the distance will be greater than B. Distance  81-9= 
<gadget id="calculator"> 81-9</gadget>
<output>72</output>
72&gt;9.   On the first roll, the probability of getting result 9 is 2/6. In this case no other alternative values for second roll which would make the distance greater than B. So probability is 0. So next estimations are:   probability of getting 81 on the first roll (2/6) * probability of getting 9 on the second roll (2/6)= 1/9   probability of getting 729 on the first roll (1/6) * probability of getting 9, 81 on the second roll (4/6) = 1/9   probability of getting 6561 on the first roll (1/6) * probability of getting 9, 81, 729 on the first roll (5/6) = 5/36   All together: 1/9 + 1/9 + 5/36 = 13/36=13/36
<result>B</result>
The “z-number” of a number x is defined as the ones digit of 2^x. Antony rolls a die with 6 sides labeled with the integers from 1 to 6, each of which has an equal probability of landing face-up. He then takes 3^z, where z is the z-number of the result of his die roll, and plots 3^z on a number line as the point A. Finally, he repeats this entire process, this time plotting the result as the point B. What is the probability that the distance between A and B is greater than the value of B?