Roll two 6-sided dice and examine their sum.
a) How long will it take to roll "snake-eyes" (a pair of ones)?
b) What is the probability of rolling a sum of 7 on the first roll?
c) What is the probability of rolling a sum of 7 on the fourth roll?
d) What is the probability of rolling a sum of 7 by the fourth roll?
e) What is the probability of it taking more than 10 rolls to roll the sum of 7?
You are once again rolling two dice. The dice have the shape of a regular tetrahedron (triangular pyramid) with sides numbered 1, 2, 3, and 4.
1. Complete the sample space chart to show the set of all possible outcomes, and the probability distribution table for the sum of your two dice. You may leave your answers in fraction form.
2. Determine the following probabilities. For full credit, use the compound probability rules to show how to compute your answer, and state whether the events are independent, dependent, mutually exclusive, or not mutually exclusive.
a. P(rolling doubles OR a sum of 5) =
b. P(rolling doubles AND a sum of 6) =
c. P(rolling doubles OR a sum of 6) =
To answer these questions, we need to analyze the possible outcomes of rolling two 6-sided dice and examine the sums.
a) To find out how long it will take to roll "snake-eyes" (a pair of ones), we can simply roll the two dice until we achieve that outcome. The number of rolls required can vary, and you need to simulate the dice rolls until the desired outcome occurs. You can use a program or an online dice roller to perform this simulation multiple times and calculate the average number of rolls needed.
b) The probability of rolling a sum of 7 on the first roll can be found by determining the number of favorable outcomes and dividing it by the total number of possible outcomes. For a sum of 7, there are six different combinations (1&6, 2&5, 3&4, 4&3, 5&2, 6&1) that yield a sum of 7. The total number of possible outcomes is 6×6 = 36 since each die has 6 possible values. Therefore, the probability is 6/36, which simplifies to 1/6.
c) To find the probability of rolling a sum of 7 on the fourth roll, we need to consider the possible outcomes in the first three rolls that didn't result in a sum of 7. The probability of not rolling a 7 on any individual roll is 30/36 (since there are 30 combinations that do not add up to 7 out of the 36 possible outcomes). Therefore, the probability of not rolling a 7 in three consecutive rolls is (30/36)×(30/36)×(30/36) = (5/6)×(5/6)×(5/6).
d) The probability of rolling a sum of 7 by the fourth roll can be found by subtracting the probability of not rolling a sum of 7 in any of the first three rolls from 1. So, it is 1 - (5/6)×(5/6)×(5/6).
e) To find the probability of it taking more than 10 rolls to roll the sum of 7, we need to consider the probabilities of not rolling a sum of 7 in each of the first 10 rolls. Since the probability of not rolling a sum of 7 on any individual roll is 30/36, the probability of not rolling a sum of 7 in 10 consecutive rolls is (30/36)^10. Therefore, the probability of it taking more than 10 rolls is 1 - (30/36)^10.
a) To roll "snake-eyes" (a pair of ones), you need to roll two dice and have both dice show the number 1. The probability of getting a 1 on a single six-sided die is 1/6. Since you are rolling two dice, you multiply the probabilities together.
The probability of rolling "snake-eyes" on any given roll is (1/6) * (1/6) = 1/36.
b) To calculate the probability of rolling a sum of 7 on the first roll, we need to determine the number of successful outcomes (rolling a sum of 7) divided by the total number of possible outcomes.
The possible outcomes when rolling two dice are 36 (6 sides on the first die times 6 sides on the second die). Out of these 36 outcomes, there are 6 possible ways to roll a sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).
The probability of rolling a sum of 7 on the first roll is 6/36, which can be simplified to 1/6 or approximately 0.167.
c) To calculate the probability of rolling a sum of 7 on the fourth roll, we need to consider the previous outcomes as well. Since we have already determined the probability of rolling a sum of 7 on the first roll, we can use that as a starting point.
On the second roll, there are 36 possible outcomes. Out of these, 6 outcomes (rolls) would result in a sum of 7 because we have already accounted for the initial roll.
On the third roll, there are also 36 possible outcomes. Similarly, 6 outcomes would result in a sum of 7.
On the fourth roll, there are again 36 possible outcomes. Out of these, 6 outcomes would result in a sum of 7.
Therefore, the probability of rolling a sum of 7 on the fourth roll is the same as the probability of rolling a sum of 7 on any of the rolls, which is 6/36, or 1/6.
d) The probability of rolling a sum of 7 by the fourth roll is the probability of rolling a sum of 7 on the first, second, third, or fourth roll.
Since the probability of rolling a sum of 7 on any roll is 1/6, and each roll is independent of the others, we can simply add the probabilities together.
The probability of rolling a sum of 7 by the fourth roll is 1/6 + 1/6 + 1/6 + 1/6 = 4/6, or approximately 0.667.
e) To calculate the probability of it taking more than 10 rolls to roll a sum of 7, we need to subtract the probability of rolling a sum of 7 in 10 or fewer rolls from 1.
The probability of rolling a sum of 7 in 10 or fewer rolls can be calculated by subtracting the probability of not rolling a sum of 7 from 1. The probability of not rolling a sum of 7 on any given roll is 1 - (6/36) = 30/36, or 5/6 since there are 30 possible outcomes that are not a sum of 7 out of 36 possible outcomes.
Therefore, the probability of rolling a sum of 7 in 10 or fewer rolls is (5/6)^10 ≈ 0.161.
Finally, the probability of it taking more than 10 rolls to roll a sum of 7 is 1 - 0.161 = 0.839, or approximately 0.839 or 83.9%.