Roll two standard dice and add the numbers. What is the probability of getting a number larger than 4 for the first time on the third roll?
There are 4 ways to roll at most 4: 11, 12, 21, 22
rolling more than 4 on 3rd roll is thus
P(2-4) * P(2-4) * (1-P(2-4)) = 4/36 * 4/36 * 32/36 = 8/729
To determine the probability of getting a number larger than 4 for the first time on the third roll, we need to calculate the probability of not getting a number larger than 4 on the first two rolls, and getting a number larger than 4 on the third roll.
Step 1: Calculate the probability of not getting a number larger than 4 on the first roll.
Out of the 6 possible outcomes, there are 2 outcomes (5 and 6) that are larger than 4. Therefore, the probability of not getting a number larger than 4 on the first roll is 4/6 or 2/3.
Step 2: Calculate the probability of not getting a number larger than 4 on the second roll.
Since the outcomes of rolling two dice are independent, we multiply the probability of not getting a number larger than 4 on the first roll (2/3) by the probability of not getting a number larger than 4 on the second roll.
The probability of not getting a number larger than 4 on the second roll is also 2/3.
Step 3: Calculate the probability of getting a number larger than 4 on the third roll.
Since we have not gotten a number larger than 4 in the first two rolls, we are left with 4 possible outcomes: (2, 3, 4, 4). Out of these 4 outcomes, 2 of them (4 and 4) are larger than 4. Therefore, the probability of getting a number larger than 4 on the third roll is 2/4 or 1/2.
Step 4: Calculate the overall probability.
Since the outcomes of rolling two dice are independent, we multiply the probabilities calculated in step 1, step 2, and step 3.
(2/3) * (2/3) * (1/2) = 4/18 = 2/9
Therefore, the probability of getting a number larger than 4 for the first time on the third roll is 2/9.
To find the probability of getting a number larger than 4 on the third roll when rolling two standard dice, we need to break down the possible outcomes.
When rolling two dice, there are a total of 36 possible outcomes because each die has 6 sides, and the possible combinations are found by multiplying the number of sides on each die together (6 * 6 = 36).
To calculate the probability of getting a number larger than 4 on the first roll, we need to count the number of outcomes that satisfy the condition. In this case, the numbers larger than 4 are 5 and 6. So, out of the 36 possible outcomes, there are 12 outcomes that satisfy this condition: (1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6), (5, 1), (5, 2), (6, 1), (6, 2). Therefore, the probability of rolling a number larger than 4 on the first roll is 12/36 or 1/3.
Now, to calculate the probability of not getting a number larger than 4 on the first and second rolls, we subtract the probability of getting a number larger than 4 on the first roll from 1 (since the sum of all possible outcomes is always 1). Therefore, the probability of not getting a number larger than 4 on the first and second rolls is 1 - 1/3 = 2/3.
To find the probability of getting a number larger than 4 for the first time on the third roll, we need to calculate the probability of not getting a number larger than 4 on the first and second rolls and then multiply it by the probability of getting a number larger than 4 on the third roll. Since these two events are independent, we can multiply their probabilities together.
The probability of not getting a number larger than 4 on the first and second rolls is 2/3. The probability of getting a number larger than 4 on the third roll is the same as the probability of getting a number larger than 4 on the first roll, which is 1/3.
Multiplying these probabilities together, we get (2/3) * (1/3) = 2/9.
Therefore, the probability of getting a number larger than 4 for the first time on the third roll is 2/9.