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
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?
