a die is rolled three times. find the chance that not all the rolls show 3 or more spots
You can get any value for the first two rolls. The last roll has to be a 1 or 2 = 2/6 = 1/3
To calculate the chance that not all the rolls show 3 or more spots, we need to find the probability of the complement event, which is the event that all three rolls show 3 or more spots.
First, let's determine the total number of possible outcomes when rolling a single die. A standard die has 6 faces, so there are 6 possible outcomes.
To find the probability of all three rolls showing 3 or more spots, we need to consider the following:
- The first roll has 4 possible outcomes (4, 5, 6).
- The second roll also has 4 possible outcomes (4, 5, 6).
- The third roll also has 4 possible outcomes (4, 5, 6).
Since each roll is independent, the probability of all three rolls showing 3 or more spots is:
P(all rolls show 3 or more spots) = (4/6) * (4/6) * (4/6) = 64/216
Now, let's find the probability of the complement event, which is the event that not all rolls show 3 or more spots:
P(not all rolls show 3 or more spots) = 1 - P(all rolls show 3 or more spots)
P(not all rolls show 3 or more spots) = 1 - 64/216 = 152/216
Therefore, the chance that not all the rolls show 3 or more spots is 152/216.
To find the chance that not all the rolls show 3 or more spots when rolling a die three times, we need to calculate the probability of the complementary event, which is the probability that all rolls show 3 or more spots.
The probability of getting 3 or more spots on a single roll of a fair six-sided die is 4/6, or 2/3.
To find the probability of all three rolls showing 3 or more spots, we multiply the probabilities of each individual roll together, since the rolls are independent events:
P(rolling 3 or more spots on first roll) = 2/3
P(rolling 3 or more spots on second roll) = 2/3
P(rolling 3 or more spots on third roll) = 2/3
P(all three rolls show 3 or more spots) = (2/3) * (2/3) * (2/3) = 8/27
Now, to find the probability that not all the rolls show 3 or more spots, we take the complement of the above probability, which is 1 - (8/27):
P(not all rolls show 3 or more spots) = 1 - 8/27 = 19/27
Therefore, there is a 19/27 chance that not all the rolls show 3 or more spots when rolling a die three times.