Mixing
Probability that a 5 occurs first
Clash Royale CLAN TAG#URR8PPP
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Suppose we roll pair of dice until a sum of either 5 or 7 appears.
What is the probability that a sum of 5 occurs first?
Try:
Let $A$ be the event that a sum of $5$ occurs on the ith roll and $B$ that the sum of 7 occurs on the ith roll. We are interested on the event $A | A^c cup B^c $. We have
$$ P(A | A^c cup B^c) = dfrac P(A cap (A^c cup B^c))P(A^c cup B^c) = fracP(A cap B^c)1 - P(A cap B) = fracP(A cap B^c)1-0 = P(A cap B^c)$$
Is this approach correct so far?
probability
asked 53 mins ago
Jimmy Sabater
1,468115
add a comment |Â
up vote
1
down vote
favorite
Suppose we roll pair of dice until a sum of either 5 or 7 appears.
What is the probability that a sum of 5 occurs first?
Try:
Let $A$ be the event that a sum of $5$ occurs on the ith roll and $B$ that the sum of 7 occurs on the ith roll. We are interested on the event $A | A^c cup B^c $. We have
$$ P(A | A^c cup B^c) = dfrac P(A cap (A^c cup B^c))P(A^c cup B^c) = fracP(A cap B^c)1 - P(A cap B) = fracP(A cap B^c)1-0 = P(A cap B^c)$$
Is this approach correct so far?
probability
asked 53 mins ago
Jimmy Sabater
1,468115
Possible duplicate of A pair of unbiased dice are rolled together till a sum of either 5 or 7 is obtained, find the probability that 5 comes before 7
– Rob
47 mins ago
Possible duplicate of Probability sum of 5 before sum of 7
– Key Flex
3 mins ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose we roll pair of dice until a sum of either 5 or 7 appears.
What is the probability that a sum of 5 occurs first?
Try:
Let $A$ be the event that a sum of $5$ occurs on the ith roll and $B$ that the sum of 7 occurs on the ith roll. We are interested on the event $A | A^c cup B^c $. We have
$$ P(A | A^c cup B^c) = dfrac P(A cap (A^c cup B^c))P(A^c cup B^c) = fracP(A cap B^c)1 - P(A cap B) = fracP(A cap B^c)1-0 = P(A cap B^c)$$
Is this approach correct so far?
probability
asked 53 mins ago
Jimmy Sabater
1,468115
Suppose we roll pair of dice until a sum of either 5 or 7 appears.
What is the probability that a sum of 5 occurs first?
Try:
Let $A$ be the event that a sum of $5$ occurs on the ith roll and $B$ that the sum of 7 occurs on the ith roll. We are interested on the event $A | A^c cup B^c $. We have
$$ P(A | A^c cup B^c) = dfrac P(A cap (A^c cup B^c))P(A^c cup B^c) = fracP(A cap B^c)1 - P(A cap B) = fracP(A cap B^c)1-0 = P(A cap B^c)$$
Is this approach correct so far?
probability
probability
asked 53 mins ago
Jimmy Sabater
1,468115
asked 53 mins ago
Jimmy Sabater
1,468115
asked 53 mins ago
Jimmy Sabater
1,468115
asked 53 mins ago
Jimmy Sabater
1,468115
asked 53 mins ago
Jimmy Sabater
1,468115
1,468115
Possible duplicate of A pair of unbiased dice are rolled together till a sum of either 5 or 7 is obtained, find the probability that 5 comes before 7
– Rob
47 mins ago
Possible duplicate of Probability sum of 5 before sum of 7
– Key Flex
3 mins ago
add a comment |Â
Possible duplicate of A pair of unbiased dice are rolled together till a sum of either 5 or 7 is obtained, find the probability that 5 comes before 7
– Rob
47 mins ago
Possible duplicate of Probability sum of 5 before sum of 7
– Key Flex
3 mins ago
Possible duplicate of A pair of unbiased dice are rolled together till a sum of either 5 or 7 is obtained, find the probability that 5 comes before 7
– Rob
47 mins ago
Possible duplicate of A pair of unbiased dice are rolled together till a sum of either 5 or 7 is obtained, find the probability that 5 comes before 7
– Rob
47 mins ago
Possible duplicate of Probability sum of 5 before sum of 7
– Key Flex
3 mins ago
Possible duplicate of Probability sum of 5 before sum of 7
– Key Flex
3 mins ago
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
1
down vote
accepted
Guide:
- Compute the probability of $7$ appears, call it $q$.
- Compute the probability of $5$ appears, call it $p$.
We are interested in
$$sum_i=1^infty (1-p-q)^i-1p$$
Remark about your attempt:
Are you intending to use some indices to denote $i$-th throw?
answered 47 mins ago
Siong Thye Goh
85.3k1457107
add a comment |Â
up vote
2
down vote
A cleaner approach would be to consider the experiment until either a sum of $5$ or $7$ is rolled at some step, say $n$. (Note that $n<infty$ almost surely.) Then, the event $A$ is that $5$ was rolled and $B$ is that $7$ was rolled, and you need
$$
mathbbP[A|A cup B]
$$
where $A$ and $B$ are disjoint...
answered 49 mins ago
gt6989b
31.2k22349
add a comment |Â
up vote
2
down vote
All events are ignored until one in the desired event space are achieved. Thus you can consider this the entire sample space.
$$S=(1,4),(4,1), (2,3), (3,2), (1,6), (6,1), (2,5), (5,2), (3,4), (4,3)$$
Now it's just the law of classical probability.
answered 42 mins ago
David Peterson
8,30121834
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Guide:
- Compute the probability of $7$ appears, call it $q$.
- Compute the probability of $5$ appears, call it $p$.
We are interested in
$$sum_i=1^infty (1-p-q)^i-1p$$
Remark about your attempt:
Are you intending to use some indices to denote $i$-th throw?
answered 47 mins ago
Siong Thye Goh
85.3k1457107
add a comment |Â
up vote
1
down vote
accepted
Guide:
- Compute the probability of $7$ appears, call it $q$.
- Compute the probability of $5$ appears, call it $p$.
We are interested in
$$sum_i=1^infty (1-p-q)^i-1p$$
Remark about your attempt:
Are you intending to use some indices to denote $i$-th throw?
answered 47 mins ago
Siong Thye Goh
85.3k1457107
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Guide:
- Compute the probability of $7$ appears, call it $q$.
- Compute the probability of $5$ appears, call it $p$.
We are interested in
$$sum_i=1^infty (1-p-q)^i-1p$$
Remark about your attempt:
Are you intending to use some indices to denote $i$-th throw?
answered 47 mins ago
Siong Thye Goh
85.3k1457107
Guide:
- Compute the probability of $7$ appears, call it $q$.
- Compute the probability of $5$ appears, call it $p$.
We are interested in
$$sum_i=1^infty (1-p-q)^i-1p$$
Remark about your attempt:
Are you intending to use some indices to denote $i$-th throw?
answered 47 mins ago
Siong Thye Goh
85.3k1457107
answered 47 mins ago
Siong Thye Goh
85.3k1457107
answered 47 mins ago
Siong Thye Goh
85.3k1457107
answered 47 mins ago
Siong Thye Goh
85.3k1457107
85.3k1457107
add a comment |Â
add a comment |Â
up vote
2
down vote
A cleaner approach would be to consider the experiment until either a sum of $5$ or $7$ is rolled at some step, say $n$. (Note that $n<infty$ almost surely.) Then, the event $A$ is that $5$ was rolled and $B$ is that $7$ was rolled, and you need
$$
mathbbP[A|A cup B]
$$
where $A$ and $B$ are disjoint...
answered 49 mins ago
gt6989b
31.2k22349
add a comment |Â
up vote
2
down vote
A cleaner approach would be to consider the experiment until either a sum of $5$ or $7$ is rolled at some step, say $n$. (Note that $n<infty$ almost surely.) Then, the event $A$ is that $5$ was rolled and $B$ is that $7$ was rolled, and you need
$$
mathbbP[A|A cup B]
$$
where $A$ and $B$ are disjoint...
answered 49 mins ago
gt6989b
31.2k22349
add a comment |Â
up vote
2
down vote
up vote
2
down vote
A cleaner approach would be to consider the experiment until either a sum of $5$ or $7$ is rolled at some step, say $n$. (Note that $n<infty$ almost surely.) Then, the event $A$ is that $5$ was rolled and $B$ is that $7$ was rolled, and you need
$$
mathbbP[A|A cup B]
$$
where $A$ and $B$ are disjoint...
answered 49 mins ago
gt6989b
31.2k22349
A cleaner approach would be to consider the experiment until either a sum of $5$ or $7$ is rolled at some step, say $n$. (Note that $n<infty$ almost surely.) Then, the event $A$ is that $5$ was rolled and $B$ is that $7$ was rolled, and you need
$$
mathbbP[A|A cup B]
$$
where $A$ and $B$ are disjoint...
answered 49 mins ago
gt6989b
31.2k22349
answered 49 mins ago
gt6989b
31.2k22349
answered 49 mins ago
gt6989b
31.2k22349
answered 49 mins ago
gt6989b
31.2k22349
31.2k22349
add a comment |Â
add a comment |Â
up vote
2
down vote
All events are ignored until one in the desired event space are achieved. Thus you can consider this the entire sample space.
$$S=(1,4),(4,1), (2,3), (3,2), (1,6), (6,1), (2,5), (5,2), (3,4), (4,3)$$
Now it's just the law of classical probability.
answered 42 mins ago
David Peterson
8,30121834
add a comment |Â
up vote
2
down vote
All events are ignored until one in the desired event space are achieved. Thus you can consider this the entire sample space.
$$S=(1,4),(4,1), (2,3), (3,2), (1,6), (6,1), (2,5), (5,2), (3,4), (4,3)$$
Now it's just the law of classical probability.
answered 42 mins ago
David Peterson
8,30121834
add a comment |Â
up vote
2
down vote
up vote
2
down vote
All events are ignored until one in the desired event space are achieved. Thus you can consider this the entire sample space.
$$S=(1,4),(4,1), (2,3), (3,2), (1,6), (6,1), (2,5), (5,2), (3,4), (4,3)$$
Now it's just the law of classical probability.
answered 42 mins ago
David Peterson
8,30121834
All events are ignored until one in the desired event space are achieved. Thus you can consider this the entire sample space.
$$S=(1,4),(4,1), (2,3), (3,2), (1,6), (6,1), (2,5), (5,2), (3,4), (4,3)$$
Now it's just the law of classical probability.
answered 42 mins ago
David Peterson
8,30121834
answered 42 mins ago
David Peterson
8,30121834
answered 42 mins ago
David Peterson
8,30121834
answered 42 mins ago
David Peterson
8,30121834
8,30121834
add a comment |Â
add a comment |Â
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Possible duplicate of A pair of unbiased dice are rolled together till a sum of either 5 or 7 is obtained, find the probability that 5 comes before 7
– Rob
47 mins ago
Possible duplicate of Probability sum of 5 before sum of 7
– Key Flex
3 mins ago