Somebody repeatedly roll a fair die. If it comes up 6 he instantly wins(and stop playing); if it comes up k, for any k between 1 and 5, he he waits k minutes and then roll again. what is the expected elapsed time from when he starts rolling until he wins?
someone answer pls!
To find the expected elapsed time from when the person starts rolling until they win, we need to calculate the expected value.
Let's denote the expected elapsed time as E, and let's consider the time spent after the first roll. There are two possibilities after the first roll:
1. The person rolls a 6 and wins immediately.
2. The person rolls a number between 1 and 5. In this case, they will wait for k minutes, and then the expected elapsed time will be E again.
The probability of rolling a 6 on the first roll is 1/6, and the expected elapsed time in this case is 0, since the person wins instantly.
The probability of rolling a number between 1 and 5 on the first roll is 5/6. In this case, the person will wait for k minutes, and then the expected elapsed time will be E. The total time spent in this case will be k minutes plus the expected elapsed time E.
Putting it all together, we can express the expected elapsed time as an equation:
E = (1/6)(0) + (5/6)(E + k)
Now, we can solve this equation to find the value of E.
Multiplying 5/6 by E + k:
E = (5/6) E + (5/6)k
Subtracting (5/6) E from both sides:
E - (5/6) E = (5/6)k
Simplifying:
(1/6) E = (5/6)k
Multiplying both sides by 6:
E = 5k
Therefore, the expected elapsed time from when the person starts rolling until they win is 5 times the number of minutes waited after rolling a number between 1 and 5.
In this scenario, since the person waits k minutes after rolling a number between 1 and 5, the expected elapsed time will be 5 times k.