Question

you roll a dice. if the result is less than 4 (excluding 4), you roll two dice and sum the results. if the result is greater than 4, you roll only one dice and use the result. what is the probability of getting a final result of 6 after this experiment?

Answer 1

To find the probability of getting a final result of 6 after this experiment, we need to calculate the probability of each scenario and then determine the probability of getting a sum of 6 when rolling two dice as well as getting a 6 when rolling one die.

Step-by-step explanation:

To find the probability of getting a final result of 6 after this experiment, we need to calculate the probability of each scenario and then determine the probability of getting a sum of 6 when rolling two dice as well as getting a 6 when rolling one die.

Scenario 1: Rolling less than 4 and rolling two dice.

P(getting a sum of 6 | rolling less than 4) = P(rolling two dice and getting a sum of 6) / P(rolling less than 4) = P(rolling two dice and getting a sum of 6) / P(rolling less than 4 and rolling two dice)

Scenario 2: Rolling greater than 4 and rolling one die.

P(getting a 6 | rolling greater than 4) = P(rolling one die and getting a 6) / P(rolling greater than 4) = P(rolling one die and getting a 6) / P(rolling greater than 4 and rolling one die)

Finally, we need to consider the probability of rolling less than 4 and rolling two dice as well as rolling greater than 4 and rolling one die.

P(final result of 6) = (P(rolling less than 4 and rolling two dice) * P(getting a sum of 6 | rolling less than 4)) + (P(rolling greater than 4 and rolling one die) * P(getting a 6 | rolling greater than 4))

Answer 2

The probability of getting a final result of 6 after the described experiment is 1/9.

Step-by-step explanation:

To calculate the probability of getting a final result of 6 after the described experiment, we need to analyze the different scenarios and calculate the probability of each.

If the initial roll of the dice is less than 4, we roll two dice and sum the results. The only way to get a sum of 6 is by rolling a 3 and a 3. The probability of rolling a 3 with a fair six-sided die is 1/6, so the probability of rolling two 3s is (1/6) x (1/6) = 1/36.

If the initial roll of the dice is greater than 4, we roll only one dice and use the result. The only way to get a 6 in this case is by rolling a 6. The probability of rolling a 6 is 1/6.

Now, let's calculate the overall probability. The probability of the initial roll being less than 4 is 3/6 (since there are three numbers less than 4).

The probability of getting a sum of two 3s from two dice rolls is 1/36. So, the probability of getting a final result of 6 from the less than 4 scenario is (3/6) x (1/36) = 1/72.

The probability of the initial roll being greater than 4 is also 3/6 (since there are three numbers greater than 4). The probability of getting a 6 from the greater than 4 scenario is 1/6. So, the probability of getting a final result of 6 from the greater than 4 scenario is (3/6) x (1/6) = 1/12.

Finally, we add together the probabilities from both scenarios to get the overall probability of getting a final result of 6: 1/72 + 1/12 = 1/9.