Kinh Nghiệm về What is the probability that the total of the two dice will add up to 7 or 11? 2022
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Nội dung chính Show- What is the probability of rolling two dice and getting a sum of 7What is the probability of rolling a 7 or 11?What is the probability of 2 dice adding up to 7?How many ways are there to roll a sum of 7 or 11 on two dice?What is the probability of getting a sum of either 11 or greater than 7 by throwing two dice?
Sample space of rolling of 2 dice is as below,
S = (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
n(S) = 36
Let A = sum of numbers is 7 = (1, 6)(2, 5)(3, 4) (4, 3) (5, 2) (6, 1)
n(A) = 6
P (Sum of numbers is 7) = n(A) / n(S)
= 6/36 = 1/6
Therefore, the probability of rolling two dice and getting a sum of 7 is 1/6.
What is the probability of rolling two dice and getting a sum of 7
Summary:
The probability of rolling two dice and getting a sum of 7 is 1/6.
You can draw up a possibility space and count how many of the outcomes meet the requirements.
#," "color(red)(7)" "," "8" "," "9" "," "10" "," "color(red)(11)" "," "color(red)(12)" ") :#
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Found 2 solutions by Greenfinch, drk:Answer by Greenfinch(383)
(Show Source): You can put this solution on YOUR website!
To do this use a Sample Space diagram. With it the problem is ridiculously simple, without it it is not all easy. Number is the addition of the values
Value of first die across top, value of second die down the side
1 2 3 4 5 6
-----------------------
1 | 2 3 4 5 6 7
2 | 3 4 5 6 7 8
3 | 4 5 6 7 8 9
4 | 5 6 7 8 9 10
5 | 6 7 8 9 10 11
6 | 7 8 9 10 11 12
There are 6 x 6 or 36 options, all are equally likely, 7 occurs 6 times, so the chances are 6/36 or 1/6. 11 occurs 2 times so chances are 2/36 or 1/18. 7 or 11 are 8 of the 36 options so 8/36 or 2/9. You can even find out chance of a prime number. the 2,3,5,7,11 options add up to 15, so 15/36 or 5/12 is the chance of
that.
Answer by drk(1908)
(Show Source): You can put this solution on YOUR website!
There are 6 ways to get a 7 and a total of 36 ways for 2 dice, so
P(7) = 6/36 = 1/6
--
There are 2 ways to get a sum of 11. So,
P(7 OR 11) = 6/36 + 2/36 = 8/36 = 2/9.
--
OR means add (+)
AND
means multiply (*)
In general, the problem of restricted partitions is quite difficult. I'll frame the problem in a more general setting:
Suppose we have $n$ dice, having $k$ faces numbered accordingly. How many ways are there to roll some positive integer $m$?
This problem can be de-worded as:
How many solutions are there to the equation $$sum_i=1^n x_i=m$$ With the condition that $x_iin mathbbN_leq k~forall iin1,...,k.$
The solution to this problem is not so simple. In small cases, like $n=2, k=6, m=7$, this can be easily checked with a table; a so called brute force approach. But for larger values of $n,k$ this is simply not feasible. Based on this post I think in general the solution to this problem is the coefficient of $x^m$ in the multinomial expansion of $$left(sum_j=1^k x^jright)^n=x^nleft(frac1-x^k1-xright)^n$$ In fact, let us define the multinomial coefficient: $$mathrmC(n,(r_1,...,r_k))=fracn!prod_j=1^k r_j!$$ And state that $$left(sum_j=1^k x_jright)^n=sum_(r_1,...,r_k)in SmathrmC(n,(r_1,...,r_k))prod_t=1^k x_t^r_t$$ Where $S$ is the set of solutions to the equation $$sum_j=1^k r_j=n$$ With the restriction that $r_jin mathbbN~forall jin1,...,k.$ However, herein lies the problem: In order to compute the number of ways to roll $m$ with $n$ $k$ sided die, which is a problem of computing restricted partitions of the number $m$, we need to find the coefficient of $x^m$ in a multinomial expansion. But, in order to compute this multinomial expansion, we need to compute restricted partitions of $n$. As you can see the problem is a bit circular. But, $n$ is usually smaller than $m$, so it might speed up the computation process a little. But the end of the day some amount of brute-force grunt work will be required.
What is the probability of rolling a 7 or 11?
Let us try to calculate the probability of winning. We can use the probabilities we calculated on the previous page. The probability of winning on the first roll is the probability of rolling 7 or 11, which is 1/6 plus 1/18, which equals to 2/9.What is the probability of 2 dice adding up to 7?
Detailed Solution. ∴ The probability of getting sum as 7 when two dice are thrown is 1/6.How many ways are there to roll a sum of 7 or 11 on two dice?
A simple but typical problem of this type: if we roll two dice, how many ways are there to get either 7 or 11? Since there are 6 ways to get 7 and two ways to get 11, the answer is 6+2=8.What is the probability of getting a sum of either 11 or greater than 7 by throwing two dice?
The probability is 25% . Tải thêm tài liệu liên quan đến nội dung bài viết What is the probability that the total of the two dice will add up to 7 or 11?