Probability of Rolling Dice

When rolling a standard six-sided die, the probability of rolling any given number is 1/6. This is because there are six equally likely outcomes, and each outcome has a probability of 1/6.

Table of Contents:

• Probability of Rolling Dice
• Formula
• Solved Problems
• FAQs

Probability of Rolling Dice

When rolling a standard six-sided die, the probability of rolling any given number is 1/6. This is because there are six equally likely outcomes, and each outcome has a probability of 1/6.

To calculate the probability of rolling a specific combination of numbers on multiple dice, you can use the multiplication rule of probability.

For example, if you roll two dice, the probability of rolling a 1 on the first die and a 2 on the second die is (1/6) * (1/6) = 1/36, since there are six possible outcomes for each die and they are independent events.

You can also calculate the probability of rolling a certain sum on multiple dice by listing all the possible outcomes and counting the number of outcomes that give the desired sum.

For example, the probability of rolling a sum of 7 on two dice is 6/36, or 1/6, since there are six possible ways to roll a sum of 7 (1+6, 2+5, 3+4, 4+3, 5+2, and 6+1) out of a total of 36 possible outcomes.

Similarly, you can calculate the probability of rolling a sum greater than or equal to a certain value by counting the number of outcomes that meet the condition and dividing by the total number of outcomes.

For example, the probability of rolling a sum of 8 or higher on two dice is 10/36, or 5/18, since there are 10 possible ways to roll a sum of 8 or higher (2+6, 3+5, 4+4, 5+3, 6+2, 3+6, 4+5, 5+4, 6+3, and 6+4) out of a total of 36 possible outcomes.

Formula of Probability

The formula for calculating the probability of an event is:

P(A) = number of favorable outcomes / total number of possible outcomes

where P(A) is the probability of event A occurring.

For example, if you roll a standard six-sided die, the probability of rolling a 1 is:

P(rolling a 1) = 1 / 6

If you roll two dice and want to calculate the probability of rolling a 1 on the first die and a 2 on the second die, you can use the multiplication rule of probability:

P(rolling a 1 and a 2) = P(rolling a 1) * P(rolling a 2 given that a 1 was rolled)

Since the events are independent, the probability of rolling a 2 given that a 1 was rolled is also 1/6, so the formula becomes:

P(rolling a 1 and a 2) = (1/6) * (1/6) = 1/36

Solved Problems of Probability of Rolling Dice

Problem 1: What is the probability of rolling a sum of 7 on two standard six-sided dice?

Solution: There are 6 possible outcomes for each die, so there are 6 x 6 = 36 total possible outcomes. To roll a sum of 7, there are 6 possible combinations: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). Therefore, the probability of rolling a sum of 7 is:

P(sum of 7) = number of favorable outcomes / total number of possible outcomes
P(sum of 7) = 6 / 36
P(sum of 7) = 1/6 or about 0.167

Problem 2: What is the probability of rolling a sum of 10 on three standard six-sided dice?

Solution: There are 6 possible outcomes for each die, so there are 6 x 6 x 6 = 216 total possible outcomes. To roll a sum of 10, there are several possible combinations: (1, 4, 5), (1, 5, 4), (2, 3, 5), (2, 4, 4), (2, 5, 3), (3, 2, 5), (3, 3, 4), (3, 4, 3), (3, 5, 2), (4, 2, 4), (4, 3, 3), (4, 4, 2), (5, 1, 4), (5, 2, 3), (5, 3, 2), and (5, 4, 1). Therefore, the probability of rolling a sum of 10 is:

P(sum of 10) = number of favorable outcomes / total number of possible outcomes
P(sum of 10) = 16 / 216
P(sum of 10) = 4/54 or about 0.074

Problem 3: What is the probability of rolling at least one 6 on three standard six-sided dice?

Solution: The probability of rolling at least one 6 on one die is 1 – the probability of not rolling a 6, which is 1 – (1/6) = 5/6. Therefore, the probability of not rolling a 6 on three dice is (5/6) x (5/6) x (5/6) = 125/216. The probability of rolling at least one 6 is the complement of this, which is 1 – 125/216 = 91/216 or about 0.421.

Problem 4: What is the probability of rolling a 6 on the first die and a 4 on the second die, given that the sum of the two dice is 10?

Solution: To roll a sum of 10, the only possible combination is (4, 6) or (6, 4). Therefore, the probability of rolling a 6 on the first die and a 4 on the second die, given that the sum of the two dice is 10, is:

P(6 on first die and 4 on second die | sum of 10) = P(6 on first die and 4 on second die and sum of 10) / P(sum of 10)
P(6 on first die and 4 on second die and sum of 10) = 1/36
P(sum of 10) = 2/36

Then probability will be 1/36 divided by 2/36 will be equal to 1/2.

Probability of Rolling Dice FAQS

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