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
- If one event is unaffected by the outcome of another event, the two events are said to be:
1. Dependent
2. Independent
3. Mutually exclusive
4. Mutually inclusive - Let $M$ be a randomly chosen non-empty subset of $S = \{1, 2, 3, \dots, 2026\}$.
Which of the following is the probability that the product of all the elements of $M$ is even? - A fair coin is tossed 6 times. The probability of getting at least 5 heads is _________________. (rounded off to two decimal places)
- Consider a random experiment of rolling a fair die. The probability of getting an odd number or a number less than 4 is ______ (Rounded off to two decimal places).
- In a null hypothesis significance test, the significance level $\alpha$ was set to 0.01. For which one or more of the following $p$ values would the null hypothesis be rejected?
