Review Three fair dice are tossed find the probability that the sum is 16
Mẹo Hướng dẫn Three fair dice are tossed find the probability that the sum is 16 2022
Bùi Thị Vân Thiện đang tìm kiếm từ khóa Three fair dice are tossed find the probability that the sum is 16 được Cập Nhật vào lúc : 2022-12-08 01:44:04 . Với phương châm chia sẻ Bí kíp về trong nội dung bài viết một cách Chi Tiết 2022. Nếu sau khi tham khảo tài liệu vẫn ko hiểu thì hoàn toàn có thể lại phản hồi ở cuối bài để Ad lý giải và hướng dẫn lại nha.When rolling two dice, distinguish between them in some way: a first one and second one, a left and a right, a red and a green, etc. Let (a,b) denote a possible outcome of rolling the two die, with a the number on the top of the first die and b the number on the top of the second die. Note that each of a and b can be any of the integers from 1 through 6. Here is a listing of all the joint possibilities for (a,b):
Nội dung chính Show- When 3 dice are rolled find the probability of getting a sum of 16?When 3 dice are rolled what is the probability of getting a sum of 15?What is the probability of 3 dice?When 3 dice are rolled find the probability of getting a sum of 8?
With the sample space now identified, formal probability theory requires that we identify the possible events. These are always subsets of the sample space, and must form a sigma-algebra. In an example such as this, where the sample space is finite because it has only 36 different outcomes, it is perhaps easiest to simply declare ALL subsets of the sample space to be possible events. That will be a sigma-algebra and avoids what might otherwise be an annoying technical difficulty. We make that declaration with this example of two dice.
With the above declaration, the outcomes where the sum of the two dice is equal to 5 form an sự kiện. If we call this sự kiện E, we have
E=(1,4),(2,3),(3,2),(4,1).Note that we have listed all the ways a first die and second die add up to 5 when we look their top faces.Consider next the probability of E, P(E). Here we need more information. If the two dice are fair and independent , each possibility (a,b) is equally likely. Because there are 36 possibilities in all, and the sum of their probabilities must equal 1, each singleton sự kiện (a,b) is assigned probability equal to 1/36. Because E is composed of 4 such distinct singleton events, P(E)=4/36= 1/9.
In general, when the two dice are fair and independent, the probability of any sự kiện is the number of elements in the sự kiện divided by 36.
What if the dice aren't fair, or aren't independent of each other? Then each outcome (a,b) is assigned a probability (a number in [0,1]) whose sum over all 36 outcomes is equal to 1. These probabilities aren't all equal, and must be estimated by experiment or inferred from other hypotheses about how the dice are related and and how likely each number is on each of the dice. Then the probability of an sự kiện such as E is the sum of the probabilities of the singleton events (a,b) that make up E.
Probability for rolling three dice with the six sided dots such as 1, 2, 3, 4, 5 and 6 dots in each (three) dies.
When three dice are thrown simultaneously/randomly, thus number of sự kiện can be 63 = (6 × 6 × 6) = 216 because each die has 1 to 6 number on its faces.Worked-out problems involving probability for rolling three dice:
1. Three dice are thrown together. Find the probability of:
(i) getting a total of 5
(ii) getting a total of atmost 5
(iii) getting a total of least 5.
(iv) getting a total of 6.
(v) getting a total of atmost 6.
(vi) getting a total of least 6.
Solution:
Three different dice are thrown the same time.
Therefore, total number of possible outcomes will be 63 = (6 × 6 × 6) = 216.(i) getting a total of 5:
Number of events of getting a total of 5 = 6
i.e. (1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1), (2, 1, 2) and (1, 2, 2)
Therefore, probability of getting a total of 5
Number of favorable outcomesP(E1) = Total number of possible outcome
= 6/216
= 1/36
(ii) getting a total of atmost 5:
Number of events of getting a total of atmost 5 = 10
i.e. (1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1) and (1, 2, 2).
Therefore, probability of getting a total of atmost 5
Number of favorable outcomesP(E2) = Total number of possible outcome
= 10/216
= 5/108
(iii) getting a total of least 5:
Number of events of getting a total of less than 5 = 4
i.e. (1, 1, 1), (1, 1, 2), (1, 2, 1) and (2, 1, 1).
Therefore, probability of getting a total of less than 5
Number of favorable outcomesP(E3) = Total number of possible outcome
= 4/216
= 1/54
Therefore, probability of getting a total of least 5 = 1 - P(getting a total of less than 5)
= 1 - 1/54
= (54 - 1)/54
= 53/54
(iv) getting a total of 6:
Number of events of getting a total of 6 = 10
i.e. (1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) and (2, 2, 2).
Therefore, probability of getting a total of 6
Number of favorable outcomesP(E4) = Total number of possible outcome
= 10/216
= 5/108
(v) getting a total of atmost 6:
Number of events of getting a total of atmost 6 = 20
i.e. (1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1), (1, 2, 2), (1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) and (2, 2, 2).
Therefore, probability of getting a total of atmost 6
Number of favorable outcomesP(E5) = Total number of possible outcome
= 20/216
= 5/54
(vi) getting a total of least 6:
Number of events of getting a total of less than 6 (sự kiện of getting a total of 3, 4 or 5) = 10
i.e. (1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1) (1, 1, 3), (1, 3, 1), (3, 1, 1), (1, 2, 2), (2, 1, 2), (2, 2, 1).
Therefore, probability of getting a total of less than 6
Number of favorable outcomesP(E6) = Total number of possible outcome
= 10/216
= 5/108
Therefore, probability of getting a total of least 6 = 1 - P(getting a total of less than 6)
= 1 - 5/108
= (108 - 5)/108
= 103/108
These examples will help us to solve different types of problems based on probability for rolling three dice.
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When 3 dice are rolled find the probability of getting a sum of 16?
Probability that the sum is 16=6/6*6*6*=1/36.When 3 dice are rolled what is the probability of getting a sum of 15?
If three dice are thrown simultaneously then probability of getting a sum of 15 is 5/108 .What is the probability of 3 dice?
The three dice are rolled fairly without any cheating. Each of the dice rolls is an Independent Event, that is the outcome from anyone dice roll has no impact whatsoever on the outcome of any other dice roll. The probability of all three happening is the product of the three probabilities: 1 × (1/6) × (1/6) = 1/36.When 3 dice are rolled find the probability of getting a sum of 8?
The total number of ways to roll an 8 with 3 dice is therefore 21, and the probability of rolling an 8 is 21/216, which is less than 5/36. Tải thêm tài liệu liên quan đến nội dung bài viết Three fair dice are tossed find the probability that the sum is 16
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