Calculating Probability of Dice Rolls
The problem asks for the probability that when rolling two identical cubic dice, at least one die shows a face value greater than 3.
A standard cubic die has faces numbered 1, 2, 3, 4, 5, 6.
- Total possible outcomes for one die: 6
- Total possible outcomes when rolling two dice: $6 \times 6 = 36$.
Using the Complement Event Approach
It's often easier to calculate the probability of the event *not* happening and subtract it from 1.
The event we want is "at least one face value > 3".
The complement event is "neither face value is greater than 3", which means "both face values are less than or equal to 3".
- Face values less than or equal to 3 are: {1, 2, 3}. There are 3 such values.
- Number of outcomes where the first die is $\le 3$: 3
- Number of outcomes where the second die is $\le 3$: 3
- Number of outcomes where *both* dice are $\le 3$: $3 \times 3 = 9$.
The probability of the complement event (both dice $\le 3$) is:
$ P(\text{both} \le 3) = \frac{\text{Number of outcomes where both} \le 3}{\text{Total possible outcomes}} = \frac{9}{36} = \frac{1}{4} $Now, we can find the probability of the original event (at least one face value > 3) using the complement rule:
$ P(\text{at least one} > 3) = 1 - P(\text{both} \le 3) $ $ P(\text{at least one} > 3) = 1 - \frac{1}{4} $ $ P(\text{at least one} > 3) = \frac{3}{4} $Final Answer
The probability that at least one of the face values is greater than 3 is $\frac{3}{4}$.
Important Questions from Probability
Three dice are thrown. What is the probability of getting a sum which is a perfect square?
Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ?
Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
What is the probability that all three boys sit together?
