Mixing
How can we define the Ordered Union of events?
Clash Royale CLAN TAG#URR8PPP
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Consider an urn $C$ which contains three (distinguishable) kinds of elements, say $alpha>0$ elements of kind $A$, $beta>0$ elements of kind $B$, and $gamma>0$ elements of kind $G$.
We perform $n>0$ trials (with replacement) of one element at a time from the urn $C$.
We define the event $L^AB_n$: "to get at least one element of kind $A$ and at least one element of kind $G$, in $n$ trials".
The probability to get a success for this event in correspondence of the trial $kin1,2ldots n$ is
$$
P(L^AB_k)=1-left(fracalpha+gammaalpha+beta+gammaright)^k-left(fracbeta+gammaalpha+beta+gammaright)^k+left(fracgammaalpha+beta+gammaright)^k.
$$
Moreover, it clearly holds the relation
$$
P(bigcup_k=1^n L_k^AB)=P(L_n^AB).
$$
The union of the events (as expressed above) naturally does not take into account the order in which they occur. But, in our case, it is not possible to get success for the event $L^AB_k$ but not for the event $L^AB_k+1$ (or any following other).
In other words, it seems to me that one should introduce the concept of ordered union of these events.
Is there any definition of such concept?
probability probability-theory
edited Jul 19 at 7:19
Asaf Karagila♦
292k31403733
asked Jul 19 at 6:53
Andrea Prunotto
578215
add a comment |Â
up vote
0
down vote
favorite
Consider an urn $C$ which contains three (distinguishable) kinds of elements, say $alpha>0$ elements of kind $A$, $beta>0$ elements of kind $B$, and $gamma>0$ elements of kind $G$.
We perform $n>0$ trials (with replacement) of one element at a time from the urn $C$.
We define the event $L^AB_n$: "to get at least one element of kind $A$ and at least one element of kind $G$, in $n$ trials".
The probability to get a success for this event in correspondence of the trial $kin1,2ldots n$ is
$$
P(L^AB_k)=1-left(fracalpha+gammaalpha+beta+gammaright)^k-left(fracbeta+gammaalpha+beta+gammaright)^k+left(fracgammaalpha+beta+gammaright)^k.
$$
Moreover, it clearly holds the relation
$$
P(bigcup_k=1^n L_k^AB)=P(L_n^AB).
$$
The union of the events (as expressed above) naturally does not take into account the order in which they occur. But, in our case, it is not possible to get success for the event $L^AB_k$ but not for the event $L^AB_k+1$ (or any following other).
In other words, it seems to me that one should introduce the concept of ordered union of these events.
Is there any definition of such concept?
probability probability-theory
edited Jul 19 at 7:19
Asaf Karagila♦
292k31403733
asked Jul 19 at 6:53
Andrea Prunotto
578215
Can you make the urn "G" (or "U", or just call it an urn), and the third kind "C"?
– Michael
Jul 19 at 7:47
I don't see a reason to define an ordered union (other than to make indexing easy). The events themselves can specify the "times" they occur, if relevant.
– Michael
Jul 19 at 7:51
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider an urn $C$ which contains three (distinguishable) kinds of elements, say $alpha>0$ elements of kind $A$, $beta>0$ elements of kind $B$, and $gamma>0$ elements of kind $G$.
We perform $n>0$ trials (with replacement) of one element at a time from the urn $C$.
We define the event $L^AB_n$: "to get at least one element of kind $A$ and at least one element of kind $G$, in $n$ trials".
The probability to get a success for this event in correspondence of the trial $kin1,2ldots n$ is
$$
P(L^AB_k)=1-left(fracalpha+gammaalpha+beta+gammaright)^k-left(fracbeta+gammaalpha+beta+gammaright)^k+left(fracgammaalpha+beta+gammaright)^k.
$$
Moreover, it clearly holds the relation
$$
P(bigcup_k=1^n L_k^AB)=P(L_n^AB).
$$
The union of the events (as expressed above) naturally does not take into account the order in which they occur. But, in our case, it is not possible to get success for the event $L^AB_k$ but not for the event $L^AB_k+1$ (or any following other).
In other words, it seems to me that one should introduce the concept of ordered union of these events.
Is there any definition of such concept?
probability probability-theory
edited Jul 19 at 7:19
Asaf Karagila♦
292k31403733
asked Jul 19 at 6:53
Andrea Prunotto
578215
Consider an urn $C$ which contains three (distinguishable) kinds of elements, say $alpha>0$ elements of kind $A$, $beta>0$ elements of kind $B$, and $gamma>0$ elements of kind $G$.
We perform $n>0$ trials (with replacement) of one element at a time from the urn $C$.
We define the event $L^AB_n$: "to get at least one element of kind $A$ and at least one element of kind $G$, in $n$ trials".
The probability to get a success for this event in correspondence of the trial $kin1,2ldots n$ is
$$
P(L^AB_k)=1-left(fracalpha+gammaalpha+beta+gammaright)^k-left(fracbeta+gammaalpha+beta+gammaright)^k+left(fracgammaalpha+beta+gammaright)^k.
$$
Moreover, it clearly holds the relation
$$
P(bigcup_k=1^n L_k^AB)=P(L_n^AB).
$$
The union of the events (as expressed above) naturally does not take into account the order in which they occur. But, in our case, it is not possible to get success for the event $L^AB_k$ but not for the event $L^AB_k+1$ (or any following other).
In other words, it seems to me that one should introduce the concept of ordered union of these events.
Is there any definition of such concept?
probability probability-theory
edited Jul 19 at 7:19
Asaf Karagila♦
292k31403733
asked Jul 19 at 6:53
Andrea Prunotto
578215
edited Jul 19 at 7:19
Asaf Karagila♦
292k31403733
edited Jul 19 at 7:19
Asaf Karagila♦
292k31403733
edited Jul 19 at 7:19
Asaf Karagila♦
292k31403733
292k31403733
asked Jul 19 at 6:53
Andrea Prunotto
578215
asked Jul 19 at 6:53
Andrea Prunotto
578215
asked Jul 19 at 6:53
Andrea Prunotto
578215
578215
Can you make the urn "G" (or "U", or just call it an urn), and the third kind "C"?
– Michael
Jul 19 at 7:47
I don't see a reason to define an ordered union (other than to make indexing easy). The events themselves can specify the "times" they occur, if relevant.
– Michael
Jul 19 at 7:51
add a comment |Â
Can you make the urn "G" (or "U", or just call it an urn), and the third kind "C"?
– Michael
Jul 19 at 7:47
I don't see a reason to define an ordered union (other than to make indexing easy). The events themselves can specify the "times" they occur, if relevant.
– Michael
Jul 19 at 7:51
Can you make the urn "G" (or "U", or just call it an urn), and the third kind "C"?
– Michael
Jul 19 at 7:47
Can you make the urn "G" (or "U", or just call it an urn), and the third kind "C"?
– Michael
Jul 19 at 7:47
I don't see a reason to define an ordered union (other than to make indexing easy). The events themselves can specify the "times" they occur, if relevant.
– Michael
Jul 19 at 7:51
I don't see a reason to define an ordered union (other than to make indexing easy). The events themselves can specify the "times" they occur, if relevant.
– Michael
Jul 19 at 7:51
add a comment |Â
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Can you make the urn "G" (or "U", or just call it an urn), and the third kind "C"?
– Michael
Jul 19 at 7:47
I don't see a reason to define an ordered union (other than to make indexing easy). The events themselves can specify the "times" they occur, if relevant.
– Michael
Jul 19 at 7:51