The problem asks to find the probability that a balanced die (standard 6-sided die) shows an even number on both tosses, when tossed twice.

Step-by-Step Solution:

1. Identify the even numbers on a die: A die has numbers 1 to 6. The even numbers are 2, 4, and 6.

   • So, the probability of getting an even number on a single toss is: $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$

2. Tossing the die twice: Since the die is tossed twice, and the events are independent, the probability of getting an even number on both tosses is the product of the probabilities for each toss. $P(\text{even on both tosses}) = P(\text{even on first toss}) \times P(\text{even on second toss})$ $P(\text{even on both tosses}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Thus, the probability that the die will show an even number on both tosses is $\frac{1}{4}$.

Would you like further details or have any questions?

Here are 5 related questions to further explore this concept:

1. What is the probability of getting an odd number on both tosses?
2. What is the probability of getting an even number on only one of the tosses?
3. What is the probability of getting a 6 on both tosses?
4. What is the probability of getting an even number at least once in two tosses?
5. If the die is tossed three times, what is the probability of getting an even number on all three tosses?

Tip: When calculating probabilities for independent events, remember to multiply the probabilities of each individual event.