Understanding Probability with Two Dice

When we toss two standard six-sided dice simultaneously, we are exploring the concept of probability. Each die can land on any number from 1 to 6. To find the probability of a specific event, we need to determine the total number of possible outcomes and the number of outcomes that satisfy our condition (favorable outcomes).

Total Possible Outcomes when Tossing Two Dice

For the first die, there are 6 possible outcomes (1, 2, 3, 4, 5, 6). For the second die, there are also 6 possible outcomes. Since the outcome of one die does not affect the outcome of the other, the total number of possible outcomes when tossing two dice simultaneously is the product of the outcomes for each die.

Total Outcomes = Outcomes on Die 1 \(\times\) Outcomes on Die 2

Total Outcomes = \(6 \times 6 = 36\)

These 36 outcomes represent all possible combinations of the numbers that can appear on the two dice. We can represent these outcomes as ordered pairs, where the first number is the result of the first die and the second number is the result of the second die.

Favorable Outcomes: Sum of 8

The question asks for the probability that the sum of the numbers appearing on both the dice is 8. We need to find all the pairs of numbers from the 36 possible outcomes whose sum is exactly 8. Let's list these pairs:

• If the first die shows 2, the second die must show 6 to get a sum of \(2+6=8\). This is the pair (2, 6).
• If the first die shows 3, the second die must show 5 to get a sum of \(3+5=8\). This is the pair (3, 5).
• If the first die shows 4, the second die must show 4 to get a sum of \(4+4=8\). This is the pair (4, 4).
• If the first die shows 5, the second die must show 3 to get a sum of \(5+3=8\). This is the pair (5, 3).
• If the first die shows 6, the second die must show 2 to get a sum of \(6+2=8\). This is the pair (6, 2).

These are the only combinations where the sum of the numbers on the two dice is 8. Counting these pairs, we find there are 5 favorable outcomes.

Favorable Outcomes = 5

Calculating the Probability

The probability of an event is calculated using the formula:

\[\text{Probability (Event)} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\]

In this case, the event is getting a sum of 8 when tossing two dice.

\[\text{Probability (Sum is 8)} = \frac{\text{Number of pairs summing to 8}}{\text{Total number of outcomes when tossing two dice}}\]

\[\text{Probability (Sum is 8)} = \frac{5}{36}\]

Therefore, the probability that the sum of the numbers appearing on both the dice is 8 is \(\frac{5}{36}\).

Summary of Probability Calculation

Understanding probability helps us predict the likelihood of different events when dealing with random processes like tossing dice. The total outcomes when tossing two dice are always 36, making the calculation of probability straightforward once favorable outcomes are identified.