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tomdemaine
10-02-2005, 08:42 AM
I just watched the Futurama episode tale of two santas and thought there might be an interesting probability question there.

Bender the lovable robot is about to be put to death. His excecution is performed when a random number generator with two digits (ie between -99 and 99) reaches 00. If the number changes every second how long on average can he expect to live?

For extra credit suppose the random number generator could hit any integer on the number line. How long would you expect to live then?

BruceZ
10-02-2005, 09:00 AM
[ QUOTE ]
I just watched the Futurama episode tale of two santas and thought there might be an interesting probability question there.

Bender the lovable robot is about to be put to death. His excecution is performed when a random number generator with two digits (ie between -99 and 99) reaches 00. If the number changes every second how long on average can he expect to live?

[/ QUOTE ]

199 seconds, if all numbers are equally likely.

My extra credit:

1. What is the most likely length of time it will take him to be executed?

2. The probability that he will be dead after N seconds is approximately 50%. What is N?

[ QUOTE ]
For extra credit suppose the random number generator could hit any integer on the number line. How long would you expect to live then?

[/ QUOTE ]

The expected value of his lifespan is infinite (undefined).

tomdemaine
10-02-2005, 12:10 PM
[ QUOTE ]
[ QUOTE ]
I just watched the Futurama episode tale of two santas and thought there might be an interesting probability question there.

Bender the lovable robot is about to be put to death. His excecution is performed when a random number generator with two digits (ie between -99 and 99) reaches 00. If the number changes every second how long on average can he expect to live?

[/ QUOTE ]

199 seconds, if all numbers are equally likely.

[/ QUOTE ]

Are you sure this is right? This means that every number but 00 will come up before 00 which doesn't seems very likely to me. Just trying to get my head aroud it. Every second there is a 1 in 199 chance of dying translates directly into 199 seconds expected lifespan?

LetYouDown
10-02-2005, 12:14 PM
[ QUOTE ]
2. The probability that he will be dead after N seconds is approximately 50%. What is N?

[/ QUOTE ]
138?

BruceZ
10-02-2005, 12:54 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
I just watched the Futurama episode tale of two santas and thought there might be an interesting probability question there.

Bender the lovable robot is about to be put to death. His excecution is performed when a random number generator with two digits (ie between -99 and 99) reaches 00. If the number changes every second how long on average can he expect to live?

[/ QUOTE ]

199 seconds, if all numbers are equally likely.

[/ QUOTE ]

Are you sure this is right? This means that every number but 00 will come up before 00 which doesn't seems very likely to me.

[/ QUOTE ]

Why should every number but 00 have to come up? The numbers are not being selected without replacement are they? I thought that every second the random number generator can output any one of the 199 numbers. Then since the probability of 00 on any second is 1 in 199, then of course the average waiting time for 00 is 199 seconds. In general, if the probability of success on a trial is p, then the average number of independent trials for a success is 1/p.

bobman0330
10-02-2005, 06:49 PM
[ QUOTE ]
1. What is the most likely length of time it will take him to be executed?

[/ QUOTE ]

1 second?

Neil Stevens
10-02-2005, 09:00 PM
Vaguely worded question, so I'll just answer this: After how many seconds do we expect him to have only a 50/50 chance of not having blown up?

(198/199)^N = 0.5

Then round up to the nearest second, I come up with 138 seconds, which is exactly what LetYouDown came up with, so I guess that's confirmation.

BruceZ
10-02-2005, 09:09 PM
[ QUOTE ]
Vaguely worded question, so I'll just answer this: After how many seconds do we expect him to have only a 50/50 chance of not having blown up?

(198/199)^N = 0.5

Then round up to the nearest second, I come up with 138 seconds, which is exactly what LetYouDown came up with, so I guess that's confirmation.

[/ QUOTE ]

That's the answer to the second question I asked, but note that this is different from the question the OP asked. He is asking for the mean, while I was asking for the median.

alThor
10-02-2005, 09:18 PM
[ QUOTE ]
For extra credit suppose the random number generator could hit any integer on the number line. How long would you expect to live then?

[/ QUOTE ]

Not to nitpick, but there is no such thing as a uniform probability distribution over the set of all integers, so the question doesn't really have an answer.

It was already noted that the distributions over integers in [-x,+x] gives you an expected waiting time around 2x. So as x goes to infinity, the expected waiting does too. That could justify the answer of "infinity".

alThor

Neil Stevens
10-02-2005, 09:20 PM
[ QUOTE ]
[ QUOTE ]
Vaguely worded question, so I'll just answer this: After how many seconds do we expect him to have only a 50/50 chance of not having blown up?

(198/199)^N = 0.5

Then round up to the nearest second, I come up with 138 seconds, which is exactly what LetYouDown came up with, so I guess that's confirmation.

[/ QUOTE ]

That's the answer to the second question I asked, but note that this is different from the question the OP asked. He is asking for the mean, while I was asking for the median.

[/ QUOTE ]I must have missed the word mean in there, heh.

BruceZ
10-02-2005, 09:35 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Vaguely worded question, so I'll just answer this: After how many seconds do we expect him to have only a 50/50 chance of not having blown up?

(198/199)^N = 0.5

Then round up to the nearest second, I come up with 138 seconds, which is exactly what LetYouDown came up with, so I guess that's confirmation.

[/ QUOTE ]

That's the answer to the second question I asked, but note that this is different from the question the OP asked. He is asking for the mean, while I was asking for the median.

[/ QUOTE ]I must have missed the word mean in there, heh.

[/ QUOTE ]

It wasn't there, but "how long on average does he expect to live" implies the expected value or mean, as long as there isn't any information to the contrary, though there are other types of averages. This is not the same as the amount of time that gives him a 50% chance of being killed, which is the median = 138 seconds. In fact, he has about a 63% chance of being killed in the mean length of time he has to live, which is 199 seconds.

Neil Stevens
10-02-2005, 10:47 PM
[ QUOTE ]
It wasn't there, but "how long on average does he expect to live" implies the expected value or mean, as long as there isn't any information to the contrary, though there are other types of averages. This is not the same as the amount of time that gives him a 50% chance of being killed, which is the median = 138 seconds. In fact, he has about a 63% chance of being killed in the mean length of time he has to live, which is 199 seconds.

[/ QUOTE ]
Guess I'm too used to non-mathematically inclined people who usually don't know exactly what they mean when they say average.

thanks,