Consider a case where two identical six-faced fair dice are rolled simultaneously. The probability of getting even numbers in both AND a sum equal to 5 and above is ______ (rounded off to two decimal places).
Dice Probability: Even Numbers and Sum
We need to find the probability of rolling two identical fair dice such that both dice show even numbers AND their sum is 5 or greater.
Sample Space Analysis
- When rolling two six-faced dice, the total number of possible outcomes is $6 \times 6 = 36$.
Favorable Outcomes Calculation
First, identify outcomes where both dice show even numbers. The even numbers are {2, 4, 6}. The possible pairs are:
- (2, 2), (2, 4), (2, 6)
- (4, 2), (4, 4), (4, 6)
- (6, 2), (6, 4), (6, 6)
There are $3 \times 3 = 9$ such outcomes.
Next, from these 9 outcomes, identify those where the sum is 5 or greater ($\text{Sum} \ge 5$):
- (2, 2): Sum = 4 (Exclude)
- (2, 4): Sum = 6 (Include)
- (2, 6): Sum = 8 (Include)
- (4, 2): Sum = 6 (Include)
- (4, 4): Sum = 8 (Include)
- (4, 6): Sum = 10 (Include)
- (6, 2): Sum = 8 (Include)
- (6, 4): Sum = 10 (Include)
- (6, 6): Sum = 12 (Include)
There are 8 outcomes that satisfy both conditions.
Probability Calculation
The probability is the ratio of favorable outcomes to the total possible outcomes.
Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{8}{36}$
Simplifying the fraction:
Probability = $\frac{8}{36} = \frac{2}{9}$
Converting the fraction to a decimal:
Probability $\approx 0.2222...$
Rounding to two decimal places, the probability is 0.22.
This value (0.22) lies between 0.21 and 0.23.
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?
