Confusing probability riddle [duplicate]
This question already has an answer here:
Probability of population to go extinct
2 answers
There exists a Snail lineage .
in each generation every snail, independent from the other:
Dies at probability 1/3,
Have a single offspring and then dies at probability 1/3
Have 2 offspring and then dies at probability 1/3
the lineage goes extinct if from the single offspring in generation 0 ,there are no remaining offspring in some generation i .
what is the probability that the lineage goes extinct?
this is so confusing to me i cant even wrap my head around it to start, please help.
probability probability-theory expected-value
edited Dec 5 '18 at 18:34
Bernard
118k639112
asked Dec 5 '18 at 17:51
user3184910user3184910
344
marked as duplicate by amd, amWhy, Lord_Farin, NCh, Lord Shark the Unknown Dec 6 '18 at 2:19
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
This question already has an answer here:
Probability of population to go extinct
2 answers
There exists a Snail lineage .
in each generation every snail, independent from the other:
Dies at probability 1/3,
Have a single offspring and then dies at probability 1/3
Have 2 offspring and then dies at probability 1/3
the lineage goes extinct if from the single offspring in generation 0 ,there are no remaining offspring in some generation i .
what is the probability that the lineage goes extinct?
this is so confusing to me i cant even wrap my head around it to start, please help.
probability probability-theory expected-value
edited Dec 5 '18 at 18:34
Bernard
118k639112
asked Dec 5 '18 at 17:51
user3184910user3184910
344
marked as duplicate by amd, amWhy, Lord_Farin, NCh, Lord Shark the Unknown Dec 6 '18 at 2:19
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
is there a limit on how many generations are we considering? If there is no limit on that then probability of extinction will be 1. (It will always extinct at some time in the future)
– MoonKnight
Dec 5 '18 at 17:58
@MoonKnight, this is not true in general. Take a look at section 2.4 of Lawler's "Introduction to Stochastic Processes" for a rigorous approach. Page 56 is what you are interested in.
– Steve L
Dec 5 '18 at 20:10
add a comment |
This question already has an answer here:
Probability of population to go extinct
2 answers
There exists a Snail lineage .
in each generation every snail, independent from the other:
Dies at probability 1/3,
Have a single offspring and then dies at probability 1/3
Have 2 offspring and then dies at probability 1/3
the lineage goes extinct if from the single offspring in generation 0 ,there are no remaining offspring in some generation i .
what is the probability that the lineage goes extinct?
this is so confusing to me i cant even wrap my head around it to start, please help.
probability probability-theory expected-value
edited Dec 5 '18 at 18:34
Bernard
118k639112
asked Dec 5 '18 at 17:51
user3184910user3184910
344
This question already has an answer here:
Probability of population to go extinct
2 answers
There exists a Snail lineage .
in each generation every snail, independent from the other:
Dies at probability 1/3,
Have a single offspring and then dies at probability 1/3
Have 2 offspring and then dies at probability 1/3
the lineage goes extinct if from the single offspring in generation 0 ,there are no remaining offspring in some generation i .
what is the probability that the lineage goes extinct?
this is so confusing to me i cant even wrap my head around it to start, please help.
This question already has an answer here:
Probability of population to go extinct
2 answers
probability probability-theory expected-value
probability probability-theory expected-value
edited Dec 5 '18 at 18:34
Bernard
118k639112
asked Dec 5 '18 at 17:51
user3184910user3184910
344
edited Dec 5 '18 at 18:34
Bernard
118k639112
asked Dec 5 '18 at 17:51
user3184910user3184910
344
edited Dec 5 '18 at 18:34
Bernard
118k639112
edited Dec 5 '18 at 18:34
Bernard
118k639112
edited Dec 5 '18 at 18:34
Bernard
118k639112
118k639112
asked Dec 5 '18 at 17:51
user3184910user3184910
344
asked Dec 5 '18 at 17:51
user3184910user3184910
344
asked Dec 5 '18 at 17:51
user3184910user3184910
344
344
marked as duplicate by amd, amWhy, Lord_Farin, NCh, Lord Shark the Unknown Dec 6 '18 at 2:19
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by amd, amWhy, Lord_Farin, NCh, Lord Shark the Unknown Dec 6 '18 at 2:19
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
is there a limit on how many generations are we considering? If there is no limit on that then probability of extinction will be 1. (It will always extinct at some time in the future)
– MoonKnight
Dec 5 '18 at 17:58
@MoonKnight, this is not true in general. Take a look at section 2.4 of Lawler's "Introduction to Stochastic Processes" for a rigorous approach. Page 56 is what you are interested in.
– Steve L
Dec 5 '18 at 20:10
add a comment |
is there a limit on how many generations are we considering? If there is no limit on that then probability of extinction will be 1. (It will always extinct at some time in the future)
– MoonKnight
Dec 5 '18 at 17:58
@MoonKnight, this is not true in general. Take a look at section 2.4 of Lawler's "Introduction to Stochastic Processes" for a rigorous approach. Page 56 is what you are interested in.
– Steve L
Dec 5 '18 at 20:10
is there a limit on how many generations are we considering? If there is no limit on that then probability of extinction will be 1. (It will always extinct at some time in the future)
– MoonKnight
Dec 5 '18 at 17:58
is there a limit on how many generations are we considering? If there is no limit on that then probability of extinction will be 1. (It will always extinct at some time in the future)
– MoonKnight
Dec 5 '18 at 17:58
@MoonKnight, this is not true in general. Take a look at section 2.4 of Lawler's "Introduction to Stochastic Processes" for a rigorous approach. Page 56 is what you are interested in.
– Steve L
Dec 5 '18 at 20:10
@MoonKnight, this is not true in general. Take a look at section 2.4 of Lawler's "Introduction to Stochastic Processes" for a rigorous approach. Page 56 is what you are interested in.
– Steve L
Dec 5 '18 at 20:10
add a comment |
2 Answers
2
active
oldest
votes
Let's go through each of the three cases for the first generation:
- With probability $frac 1 3$ the first snake dies without any offspring, which makes the lineage go extinct.
- With probability $frac 1 3$ the first snake dies after having a single offspring. We are then left with one generation consisting of one snake, which is the same scenario we started with. Hence, if we denote by $P$ the probability that our lineage goes extinct (what we're trying to solve for), in this case the lineage will go extinct with probability $P$.
- With probability $frac 1 3$ the first snake has two offsprings and then dies. In order for this lineage to go extinct, both of the offspring's lineages need to go extinct. Each of these events happens with probability $P$, which gives a probability $P^2$ of both events happening at once.
Hence, in total, we get that $P = frac 1 3 + frac 1 3P + frac 1 3 P^2$. Solving for $P$, we get
begin{alignat*}{2}
&&3P &= 1 + P + P^2\
&iff&qquad P^2 - 2P + 1 &= 0\
&iff&qquad (P - 1)^2 &= 0\
&iff&qquad P &= 1
&end{alignat*}
so for time tending to infinity the lineage is certain to go extinct.
(There are a few issues with this, since we assume that such a probability $P$ exists in the first place, but an approach like this is nice for getting a feeling for the problem)
answered Dec 5 '18 at 18:03
arch1t3cht30arch1t3cht30
515211
add a comment |
Hint: the probability $p(n)$ that there are no descendants of a given individual after $n$ generations is $frac{1}{3} + frac{1}{3}p(n-1) + frac{1}{3}p(n-1)^2$.
answered Dec 5 '18 at 17:59
rogerlrogerl
17.4k22746
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Let's go through each of the three cases for the first generation:
- With probability $frac 1 3$ the first snake dies without any offspring, which makes the lineage go extinct.
- With probability $frac 1 3$ the first snake dies after having a single offspring. We are then left with one generation consisting of one snake, which is the same scenario we started with. Hence, if we denote by $P$ the probability that our lineage goes extinct (what we're trying to solve for), in this case the lineage will go extinct with probability $P$.
- With probability $frac 1 3$ the first snake has two offsprings and then dies. In order for this lineage to go extinct, both of the offspring's lineages need to go extinct. Each of these events happens with probability $P$, which gives a probability $P^2$ of both events happening at once.
Hence, in total, we get that $P = frac 1 3 + frac 1 3P + frac 1 3 P^2$. Solving for $P$, we get
begin{alignat*}{2}
&&3P &= 1 + P + P^2\
&iff&qquad P^2 - 2P + 1 &= 0\
&iff&qquad (P - 1)^2 &= 0\
&iff&qquad P &= 1
&end{alignat*}
so for time tending to infinity the lineage is certain to go extinct.
(There are a few issues with this, since we assume that such a probability $P$ exists in the first place, but an approach like this is nice for getting a feeling for the problem)
answered Dec 5 '18 at 18:03
arch1t3cht30arch1t3cht30
515211
add a comment |
Let's go through each of the three cases for the first generation:
- With probability $frac 1 3$ the first snake dies without any offspring, which makes the lineage go extinct.
- With probability $frac 1 3$ the first snake dies after having a single offspring. We are then left with one generation consisting of one snake, which is the same scenario we started with. Hence, if we denote by $P$ the probability that our lineage goes extinct (what we're trying to solve for), in this case the lineage will go extinct with probability $P$.
- With probability $frac 1 3$ the first snake has two offsprings and then dies. In order for this lineage to go extinct, both of the offspring's lineages need to go extinct. Each of these events happens with probability $P$, which gives a probability $P^2$ of both events happening at once.
Hence, in total, we get that $P = frac 1 3 + frac 1 3P + frac 1 3 P^2$. Solving for $P$, we get
begin{alignat*}{2}
&&3P &= 1 + P + P^2\
&iff&qquad P^2 - 2P + 1 &= 0\
&iff&qquad (P - 1)^2 &= 0\
&iff&qquad P &= 1
&end{alignat*}
so for time tending to infinity the lineage is certain to go extinct.
(There are a few issues with this, since we assume that such a probability $P$ exists in the first place, but an approach like this is nice for getting a feeling for the problem)
answered Dec 5 '18 at 18:03
arch1t3cht30arch1t3cht30
515211
add a comment |
Let's go through each of the three cases for the first generation:
- With probability $frac 1 3$ the first snake dies without any offspring, which makes the lineage go extinct.
- With probability $frac 1 3$ the first snake dies after having a single offspring. We are then left with one generation consisting of one snake, which is the same scenario we started with. Hence, if we denote by $P$ the probability that our lineage goes extinct (what we're trying to solve for), in this case the lineage will go extinct with probability $P$.
- With probability $frac 1 3$ the first snake has two offsprings and then dies. In order for this lineage to go extinct, both of the offspring's lineages need to go extinct. Each of these events happens with probability $P$, which gives a probability $P^2$ of both events happening at once.
Hence, in total, we get that $P = frac 1 3 + frac 1 3P + frac 1 3 P^2$. Solving for $P$, we get
begin{alignat*}{2}
&&3P &= 1 + P + P^2\
&iff&qquad P^2 - 2P + 1 &= 0\
&iff&qquad (P - 1)^2 &= 0\
&iff&qquad P &= 1
&end{alignat*}
so for time tending to infinity the lineage is certain to go extinct.
(There are a few issues with this, since we assume that such a probability $P$ exists in the first place, but an approach like this is nice for getting a feeling for the problem)
answered Dec 5 '18 at 18:03
arch1t3cht30arch1t3cht30
515211
Let's go through each of the three cases for the first generation:
- With probability $frac 1 3$ the first snake dies without any offspring, which makes the lineage go extinct.
- With probability $frac 1 3$ the first snake dies after having a single offspring. We are then left with one generation consisting of one snake, which is the same scenario we started with. Hence, if we denote by $P$ the probability that our lineage goes extinct (what we're trying to solve for), in this case the lineage will go extinct with probability $P$.
- With probability $frac 1 3$ the first snake has two offsprings and then dies. In order for this lineage to go extinct, both of the offspring's lineages need to go extinct. Each of these events happens with probability $P$, which gives a probability $P^2$ of both events happening at once.
Hence, in total, we get that $P = frac 1 3 + frac 1 3P + frac 1 3 P^2$. Solving for $P$, we get
begin{alignat*}{2}
&&3P &= 1 + P + P^2\
&iff&qquad P^2 - 2P + 1 &= 0\
&iff&qquad (P - 1)^2 &= 0\
&iff&qquad P &= 1
&end{alignat*}
so for time tending to infinity the lineage is certain to go extinct.
(There are a few issues with this, since we assume that such a probability $P$ exists in the first place, but an approach like this is nice for getting a feeling for the problem)
answered Dec 5 '18 at 18:03
arch1t3cht30arch1t3cht30
515211
answered Dec 5 '18 at 18:03
arch1t3cht30arch1t3cht30
515211
answered Dec 5 '18 at 18:03
arch1t3cht30arch1t3cht30
515211
answered Dec 5 '18 at 18:03
arch1t3cht30arch1t3cht30
515211
515211
add a comment |
add a comment |
Hint: the probability $p(n)$ that there are no descendants of a given individual after $n$ generations is $frac{1}{3} + frac{1}{3}p(n-1) + frac{1}{3}p(n-1)^2$.
answered Dec 5 '18 at 17:59
rogerlrogerl
17.4k22746
add a comment |
Hint: the probability $p(n)$ that there are no descendants of a given individual after $n$ generations is $frac{1}{3} + frac{1}{3}p(n-1) + frac{1}{3}p(n-1)^2$.
answered Dec 5 '18 at 17:59
rogerlrogerl
17.4k22746
add a comment |
Hint: the probability $p(n)$ that there are no descendants of a given individual after $n$ generations is $frac{1}{3} + frac{1}{3}p(n-1) + frac{1}{3}p(n-1)^2$.
answered Dec 5 '18 at 17:59
rogerlrogerl
17.4k22746
Hint: the probability $p(n)$ that there are no descendants of a given individual after $n$ generations is $frac{1}{3} + frac{1}{3}p(n-1) + frac{1}{3}p(n-1)^2$.
answered Dec 5 '18 at 17:59
rogerlrogerl
17.4k22746
answered Dec 5 '18 at 17:59
rogerlrogerl
17.4k22746
answered Dec 5 '18 at 17:59
rogerlrogerl
17.4k22746
answered Dec 5 '18 at 17:59
rogerlrogerl
17.4k22746
17.4k22746
add a comment |
add a comment |
is there a limit on how many generations are we considering? If there is no limit on that then probability of extinction will be 1. (It will always extinct at some time in the future)
– MoonKnight
Dec 5 '18 at 17:58
@MoonKnight, this is not true in general. Take a look at section 2.4 of Lawler's "Introduction to Stochastic Processes" for a rigorous approach. Page 56 is what you are interested in.
– Steve L
Dec 5 '18 at 20:10