A box has $4$ red and $20$ white balls. A person takes $10$ balls. What is the probability that all or none of the red balls were taken?Box A has 3 red and 7 white ballsWe have two boxes, in box #1 there are 7 white balls and 5 red balls…probabilityA box contains 5 blue balls, 4 red balls and 9 green balls. A person takes out 3 balls from the box.An urn has 3 red balls 4 white balls and 2 black ballsConditional probability on red and white ballsBoth balls are red. What is the probability that they both came from the same box?Suppose there are 9 white balls and 3 red balls. A,B,C take turns to pick a ball without replacement . The first person to get a red ball wins.What is the probability that the person picks up all three red balls?drawing 2 red balls from box contains 3 red balls and 7 white ballsA box contains 3 red balls and 2 white balls
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A box has $4$ red and $20$ white balls. A person takes $10$ balls. What is the probability that all or none of the red balls were taken?
Box A has 3 red and 7 white ballsWe have two boxes, in box #1 there are 7 white balls and 5 red balls…probabilityA box contains 5 blue balls, 4 red balls and 9 green balls. A person takes out 3 balls from the box.An urn has 3 red balls 4 white balls and 2 black ballsConditional probability on red and white ballsBoth balls are red. What is the probability that they both came from the same box?Suppose there are 9 white balls and 3 red balls. A,B,C take turns to pick a ball without replacement . The first person to get a red ball wins.What is the probability that the person picks up all three red balls?drawing 2 red balls from box contains 3 red balls and 7 white ballsA box contains 3 red balls and 2 white balls
$begingroup$
A box has $24$ balls, $4$ red and $20$ white. One person takes $10$ balls and the second the remaining $14$. What is the probability that one of the two people picked up the $4$ red ones?
I don't understand why is this correct.
$$frac20 choose 6+20 choose 1024 choose 10 $$
probability combinatorics
$endgroup$
add a comment |
$begingroup$
A box has $24$ balls, $4$ red and $20$ white. One person takes $10$ balls and the second the remaining $14$. What is the probability that one of the two people picked up the $4$ red ones?
I don't understand why is this correct.
$$frac20 choose 6+20 choose 1024 choose 10 $$
probability combinatorics
$endgroup$
add a comment |
$begingroup$
A box has $24$ balls, $4$ red and $20$ white. One person takes $10$ balls and the second the remaining $14$. What is the probability that one of the two people picked up the $4$ red ones?
I don't understand why is this correct.
$$frac20 choose 6+20 choose 1024 choose 10 $$
probability combinatorics
$endgroup$
A box has $24$ balls, $4$ red and $20$ white. One person takes $10$ balls and the second the remaining $14$. What is the probability that one of the two people picked up the $4$ red ones?
I don't understand why is this correct.
$$frac20 choose 6+20 choose 1024 choose 10 $$
probability combinatorics
probability combinatorics
edited yesterday
user
5,81311031
5,81311031
asked yesterday
GuidoGuido
355
355
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4 Answers
4
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votes
$begingroup$
Because $tfracbinom 44binom206binom1414binom2410binom1414$ is the probability that the first person select 4 red and 6 white and the second person receives the remaining 14 white, while $tfracbinom2010binom44binom1010binom2410binom1414$ is the probability that the first person selects 10 white and second person select 4 red and 10 white.
$$~\dfracbinom 44binom206binom1414+binom2010binom44binom1010binom2410binom1414=dfracbinom206+binom2010binom2410$$
$endgroup$
1
$begingroup$
I would assume that $binom204$ in the final expression is a typo...
$endgroup$
– user
yesterday
$begingroup$
Yas. Good catch.
$endgroup$
– Graham Kemp
yesterday
add a comment |
$begingroup$
You can use the Hypergeometric distribution.
a) The probability that the first person pick up 4 red ones is $fracbinom44cdot binom206binom2410$
Then the (conditional) probability that the second person does not pick up 4 red ones is just $1.$
b) The probability that the first person does not pick up 4 red ones is $fracbinom40cdot binom2010binom2410$
The (conditional) probability that the second person pick up 4 red one is $fracbinom44cdot binom1010binom1414=1$
So in total we have
$$fracbinom44cdot binom206binom2410+fracbinom40cdot binom2010binom2410=frac binom206binom2410+frac binom2010binom2410=fracbinom206+ binom2010binom2410$$
$endgroup$
add a comment |
$begingroup$
There are two ways for a "good event" to occur in this probability space: either player one picks up all 4 red balls (first type of good event), or player 1 picks up none of the red balls (second type of good event; equivalent to saying player 2 picks up all red balls).
The total number of ways that the first type of good event can occur is the number of ways we can pick 10 - 4 = 6 balls amond 24 - 4 = 20 balls after we fix that player one has selected the 4 red balls. In other words, there are $20 choose 6$ ways the first type of good event can occur.
The total number of ways that the second type of good event may occur is the number of ways player one can pick all of his balls exclusively from the white balls. Since there are 20 white balls and player 1 picks 10 balls in total, this gives us $20 choose 10$ ways for the second type of good event to occur.
Since the two "good" event spaces are mutually exclusive (player 1 can't pick all the red balls and none of the red balls at the same time), we can add them up obtain the total number of good event (the numerator in the solution you provided), and divide this by the size of the probability space (the total number of ways player 1 can pick 10 balls from 24, which is your denominator term).
Does this answer your question?
New contributor
$endgroup$
add a comment |
$begingroup$
Firstly, note that the events are mutually exclusive events. Therefore, the probability overall is the sum of the probability of the first person and second person selecting 4 red balls.
The number of ways the first person selects the 4 red balls is the same as the number of ways they can select the remaining six as white balls (they have 10 chances in total) from the 20 white balls available. This can be done in $24-4choose6$ ways
The number of ways the second person selects the 4 red balls is the same as the number of ways they can select the remaining ten as white balls (they have 14 chances on total) from the 20 white balls available. This can be done in $24-4choose10$ ways.
The total number of ways to split the 24 balls is $24choose10$
Therefore, the probability is what you put down.
New contributor
$endgroup$
add a comment |
Your Answer
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4 Answers
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4 Answers
4
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$begingroup$
Because $tfracbinom 44binom206binom1414binom2410binom1414$ is the probability that the first person select 4 red and 6 white and the second person receives the remaining 14 white, while $tfracbinom2010binom44binom1010binom2410binom1414$ is the probability that the first person selects 10 white and second person select 4 red and 10 white.
$$~\dfracbinom 44binom206binom1414+binom2010binom44binom1010binom2410binom1414=dfracbinom206+binom2010binom2410$$
$endgroup$
1
$begingroup$
I would assume that $binom204$ in the final expression is a typo...
$endgroup$
– user
yesterday
$begingroup$
Yas. Good catch.
$endgroup$
– Graham Kemp
yesterday
add a comment |
$begingroup$
Because $tfracbinom 44binom206binom1414binom2410binom1414$ is the probability that the first person select 4 red and 6 white and the second person receives the remaining 14 white, while $tfracbinom2010binom44binom1010binom2410binom1414$ is the probability that the first person selects 10 white and second person select 4 red and 10 white.
$$~\dfracbinom 44binom206binom1414+binom2010binom44binom1010binom2410binom1414=dfracbinom206+binom2010binom2410$$
$endgroup$
1
$begingroup$
I would assume that $binom204$ in the final expression is a typo...
$endgroup$
– user
yesterday
$begingroup$
Yas. Good catch.
$endgroup$
– Graham Kemp
yesterday
add a comment |
$begingroup$
Because $tfracbinom 44binom206binom1414binom2410binom1414$ is the probability that the first person select 4 red and 6 white and the second person receives the remaining 14 white, while $tfracbinom2010binom44binom1010binom2410binom1414$ is the probability that the first person selects 10 white and second person select 4 red and 10 white.
$$~\dfracbinom 44binom206binom1414+binom2010binom44binom1010binom2410binom1414=dfracbinom206+binom2010binom2410$$
$endgroup$
Because $tfracbinom 44binom206binom1414binom2410binom1414$ is the probability that the first person select 4 red and 6 white and the second person receives the remaining 14 white, while $tfracbinom2010binom44binom1010binom2410binom1414$ is the probability that the first person selects 10 white and second person select 4 red and 10 white.
$$~\dfracbinom 44binom206binom1414+binom2010binom44binom1010binom2410binom1414=dfracbinom206+binom2010binom2410$$
edited yesterday
answered yesterday
Graham KempGraham Kemp
87.2k43579
87.2k43579
1
$begingroup$
I would assume that $binom204$ in the final expression is a typo...
$endgroup$
– user
yesterday
$begingroup$
Yas. Good catch.
$endgroup$
– Graham Kemp
yesterday
add a comment |
1
$begingroup$
I would assume that $binom204$ in the final expression is a typo...
$endgroup$
– user
yesterday
$begingroup$
Yas. Good catch.
$endgroup$
– Graham Kemp
yesterday
1
1
$begingroup$
I would assume that $binom204$ in the final expression is a typo...
$endgroup$
– user
yesterday
$begingroup$
I would assume that $binom204$ in the final expression is a typo...
$endgroup$
– user
yesterday
$begingroup$
Yas. Good catch.
$endgroup$
– Graham Kemp
yesterday
$begingroup$
Yas. Good catch.
$endgroup$
– Graham Kemp
yesterday
add a comment |
$begingroup$
You can use the Hypergeometric distribution.
a) The probability that the first person pick up 4 red ones is $fracbinom44cdot binom206binom2410$
Then the (conditional) probability that the second person does not pick up 4 red ones is just $1.$
b) The probability that the first person does not pick up 4 red ones is $fracbinom40cdot binom2010binom2410$
The (conditional) probability that the second person pick up 4 red one is $fracbinom44cdot binom1010binom1414=1$
So in total we have
$$fracbinom44cdot binom206binom2410+fracbinom40cdot binom2010binom2410=frac binom206binom2410+frac binom2010binom2410=fracbinom206+ binom2010binom2410$$
$endgroup$
add a comment |
$begingroup$
You can use the Hypergeometric distribution.
a) The probability that the first person pick up 4 red ones is $fracbinom44cdot binom206binom2410$
Then the (conditional) probability that the second person does not pick up 4 red ones is just $1.$
b) The probability that the first person does not pick up 4 red ones is $fracbinom40cdot binom2010binom2410$
The (conditional) probability that the second person pick up 4 red one is $fracbinom44cdot binom1010binom1414=1$
So in total we have
$$fracbinom44cdot binom206binom2410+fracbinom40cdot binom2010binom2410=frac binom206binom2410+frac binom2010binom2410=fracbinom206+ binom2010binom2410$$
$endgroup$
add a comment |
$begingroup$
You can use the Hypergeometric distribution.
a) The probability that the first person pick up 4 red ones is $fracbinom44cdot binom206binom2410$
Then the (conditional) probability that the second person does not pick up 4 red ones is just $1.$
b) The probability that the first person does not pick up 4 red ones is $fracbinom40cdot binom2010binom2410$
The (conditional) probability that the second person pick up 4 red one is $fracbinom44cdot binom1010binom1414=1$
So in total we have
$$fracbinom44cdot binom206binom2410+fracbinom40cdot binom2010binom2410=frac binom206binom2410+frac binom2010binom2410=fracbinom206+ binom2010binom2410$$
$endgroup$
You can use the Hypergeometric distribution.
a) The probability that the first person pick up 4 red ones is $fracbinom44cdot binom206binom2410$
Then the (conditional) probability that the second person does not pick up 4 red ones is just $1.$
b) The probability that the first person does not pick up 4 red ones is $fracbinom40cdot binom2010binom2410$
The (conditional) probability that the second person pick up 4 red one is $fracbinom44cdot binom1010binom1414=1$
So in total we have
$$fracbinom44cdot binom206binom2410+fracbinom40cdot binom2010binom2410=frac binom206binom2410+frac binom2010binom2410=fracbinom206+ binom2010binom2410$$
answered yesterday
callculuscallculus
18.5k31428
18.5k31428
add a comment |
add a comment |
$begingroup$
There are two ways for a "good event" to occur in this probability space: either player one picks up all 4 red balls (first type of good event), or player 1 picks up none of the red balls (second type of good event; equivalent to saying player 2 picks up all red balls).
The total number of ways that the first type of good event can occur is the number of ways we can pick 10 - 4 = 6 balls amond 24 - 4 = 20 balls after we fix that player one has selected the 4 red balls. In other words, there are $20 choose 6$ ways the first type of good event can occur.
The total number of ways that the second type of good event may occur is the number of ways player one can pick all of his balls exclusively from the white balls. Since there are 20 white balls and player 1 picks 10 balls in total, this gives us $20 choose 10$ ways for the second type of good event to occur.
Since the two "good" event spaces are mutually exclusive (player 1 can't pick all the red balls and none of the red balls at the same time), we can add them up obtain the total number of good event (the numerator in the solution you provided), and divide this by the size of the probability space (the total number of ways player 1 can pick 10 balls from 24, which is your denominator term).
Does this answer your question?
New contributor
$endgroup$
add a comment |
$begingroup$
There are two ways for a "good event" to occur in this probability space: either player one picks up all 4 red balls (first type of good event), or player 1 picks up none of the red balls (second type of good event; equivalent to saying player 2 picks up all red balls).
The total number of ways that the first type of good event can occur is the number of ways we can pick 10 - 4 = 6 balls amond 24 - 4 = 20 balls after we fix that player one has selected the 4 red balls. In other words, there are $20 choose 6$ ways the first type of good event can occur.
The total number of ways that the second type of good event may occur is the number of ways player one can pick all of his balls exclusively from the white balls. Since there are 20 white balls and player 1 picks 10 balls in total, this gives us $20 choose 10$ ways for the second type of good event to occur.
Since the two "good" event spaces are mutually exclusive (player 1 can't pick all the red balls and none of the red balls at the same time), we can add them up obtain the total number of good event (the numerator in the solution you provided), and divide this by the size of the probability space (the total number of ways player 1 can pick 10 balls from 24, which is your denominator term).
Does this answer your question?
New contributor
$endgroup$
add a comment |
$begingroup$
There are two ways for a "good event" to occur in this probability space: either player one picks up all 4 red balls (first type of good event), or player 1 picks up none of the red balls (second type of good event; equivalent to saying player 2 picks up all red balls).
The total number of ways that the first type of good event can occur is the number of ways we can pick 10 - 4 = 6 balls amond 24 - 4 = 20 balls after we fix that player one has selected the 4 red balls. In other words, there are $20 choose 6$ ways the first type of good event can occur.
The total number of ways that the second type of good event may occur is the number of ways player one can pick all of his balls exclusively from the white balls. Since there are 20 white balls and player 1 picks 10 balls in total, this gives us $20 choose 10$ ways for the second type of good event to occur.
Since the two "good" event spaces are mutually exclusive (player 1 can't pick all the red balls and none of the red balls at the same time), we can add them up obtain the total number of good event (the numerator in the solution you provided), and divide this by the size of the probability space (the total number of ways player 1 can pick 10 balls from 24, which is your denominator term).
Does this answer your question?
New contributor
$endgroup$
There are two ways for a "good event" to occur in this probability space: either player one picks up all 4 red balls (first type of good event), or player 1 picks up none of the red balls (second type of good event; equivalent to saying player 2 picks up all red balls).
The total number of ways that the first type of good event can occur is the number of ways we can pick 10 - 4 = 6 balls amond 24 - 4 = 20 balls after we fix that player one has selected the 4 red balls. In other words, there are $20 choose 6$ ways the first type of good event can occur.
The total number of ways that the second type of good event may occur is the number of ways player one can pick all of his balls exclusively from the white balls. Since there are 20 white balls and player 1 picks 10 balls in total, this gives us $20 choose 10$ ways for the second type of good event to occur.
Since the two "good" event spaces are mutually exclusive (player 1 can't pick all the red balls and none of the red balls at the same time), we can add them up obtain the total number of good event (the numerator in the solution you provided), and divide this by the size of the probability space (the total number of ways player 1 can pick 10 balls from 24, which is your denominator term).
Does this answer your question?
New contributor
New contributor
answered yesterday
ChrisHansenChrisHansen
111
111
New contributor
New contributor
add a comment |
add a comment |
$begingroup$
Firstly, note that the events are mutually exclusive events. Therefore, the probability overall is the sum of the probability of the first person and second person selecting 4 red balls.
The number of ways the first person selects the 4 red balls is the same as the number of ways they can select the remaining six as white balls (they have 10 chances in total) from the 20 white balls available. This can be done in $24-4choose6$ ways
The number of ways the second person selects the 4 red balls is the same as the number of ways they can select the remaining ten as white balls (they have 14 chances on total) from the 20 white balls available. This can be done in $24-4choose10$ ways.
The total number of ways to split the 24 balls is $24choose10$
Therefore, the probability is what you put down.
New contributor
$endgroup$
add a comment |
$begingroup$
Firstly, note that the events are mutually exclusive events. Therefore, the probability overall is the sum of the probability of the first person and second person selecting 4 red balls.
The number of ways the first person selects the 4 red balls is the same as the number of ways they can select the remaining six as white balls (they have 10 chances in total) from the 20 white balls available. This can be done in $24-4choose6$ ways
The number of ways the second person selects the 4 red balls is the same as the number of ways they can select the remaining ten as white balls (they have 14 chances on total) from the 20 white balls available. This can be done in $24-4choose10$ ways.
The total number of ways to split the 24 balls is $24choose10$
Therefore, the probability is what you put down.
New contributor
$endgroup$
add a comment |
$begingroup$
Firstly, note that the events are mutually exclusive events. Therefore, the probability overall is the sum of the probability of the first person and second person selecting 4 red balls.
The number of ways the first person selects the 4 red balls is the same as the number of ways they can select the remaining six as white balls (they have 10 chances in total) from the 20 white balls available. This can be done in $24-4choose6$ ways
The number of ways the second person selects the 4 red balls is the same as the number of ways they can select the remaining ten as white balls (they have 14 chances on total) from the 20 white balls available. This can be done in $24-4choose10$ ways.
The total number of ways to split the 24 balls is $24choose10$
Therefore, the probability is what you put down.
New contributor
$endgroup$
Firstly, note that the events are mutually exclusive events. Therefore, the probability overall is the sum of the probability of the first person and second person selecting 4 red balls.
The number of ways the first person selects the 4 red balls is the same as the number of ways they can select the remaining six as white balls (they have 10 chances in total) from the 20 white balls available. This can be done in $24-4choose6$ ways
The number of ways the second person selects the 4 red balls is the same as the number of ways they can select the remaining ten as white balls (they have 14 chances on total) from the 20 white balls available. This can be done in $24-4choose10$ ways.
The total number of ways to split the 24 balls is $24choose10$
Therefore, the probability is what you put down.
New contributor
edited yesterday
New contributor
answered yesterday
amanaman
927
927
New contributor
New contributor
add a comment |
add a comment |
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