Four dice are rolled.
5/648
To solve this problem, we need to calculate the probability that the sum of the numbers on four dice is equal to 6.
Each die has 6 faces, numbered from 1 to 6. When four dice are rolled, there are a total of \(6^4\) possible outcomes. This is because each die operates independently and can land on any of one six faces.
Therefore, the total number of outcomes from rolling four dice is:
\(6^4 = 1296\)
Next, we determine the number of favorable outcomes, i.e., the combinations where the sum equals 6.
The possible ways to get the sum of 6 when four numbers (from 1 to 6) are added is important because each die can contribute to the total in various ways, but we need to ensure that their total is exactly 6. The outcomes can be visualized as:
- (1, 1, 1, 3)
- (1, 1, 2, 2)
- (1, 2, 1, 2)
- (1, 2, 2, 1)
- (2, 1, 1, 2)
- (2, 1, 2, 1)
- (2, 2, 1, 1)
- (3, 1, 1, 1)
Notice that these combinations need to include permutations. Let's calculate the number of permutations for each favorable outcome:
- (1, 1, 1, 3) - 4 permutations
- (1, 1, 2, 2) - 6 permutations
- (1, 2, 1, 2) - 6 permutations (similar to (1, 1, 2, 2))
- (2, 2, 1, 1) - 6 permutations (similar to (1, 1, 2, 2))
- (3, 1, 1, 1) - 4 permutations
Therefore, the total number of favorable permutations equals:
\(4 + 6 + 6 + 6 + 4 = 26\)
Thus, the probability of getting a total of 6 is:
\(\frac{26}{1296} = \frac{2}{648} = \frac{5}{648}\)
This matches the correct answer, which is \(\frac{5}{648}\).
Similar Questions
- An edible oil is sold at the rates 150, 200, 250, 300 rupees per litre in four consecutive years. Assuming that an equal amount of money is spent on oil by a family in every year during these years, what is the average price of oil in rupees (approximately) per litre ?
- Let \(m = 77^n\). The index \(n\) is given a positive integral value at random. What is the probability that the value of \(m\) will have 1 in the units place ?
- What is \(P(A) + P(B) + P(C)\) equal to ?
- A, B, C are three mutually exclusive and exhaustive events associated with a random experiment. If \(3P(B) = 4P(A)\) and \(3P(C) = 2P(B)\), then what is \(P(A)\) equal to?
- A die has two faces with number 4, three faces with number 5 and one face with number 6. If the die is rolled once, then what is the probability of getting 4 or 5?
- A box contains 2 black, 4 yellow and 6 white balls. Three balls are drawn in succession with replacement. What is the probability that all three are of the same colour?
- The letters of the word ZOOLOGY are arranged in all possible ways. What is the probability that the consonants and vowels occur alternatively?
- A natural number \(x\) is chosen at random from the first 100 natural numbers. What is the probability that \(x^2 + x > 50\)?
- A fair coin is tossed till two heads occur in succession. What is the probability that the number of tosses required is less than 6?
- Two perfect dice are thrown. What is the probability that the sum of the numbers on the faces is neither 9 nor 10?
Important Questions from Probability
Two dice are thrown simultaneously. What is the probability of getting the same number on both the dice?
The probability of being 53 Sundays in year 2020 is-
Three dice are thrown randomly. The probability of coming 3 in at least one die is
The probability of having 53 Tuesdays in an ordinary year is:
When two dice are tossed simultaneously, the probability that the sum of the numbers appearing on both the dice is 8 will be
