Two dice are thrown simultaneously. What is the probability of getting the same number on both the dice?
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Understanding Probability with Two Dice
The question asks for the probability of getting the same number on both dice when two dice are thrown simultaneously. To solve this, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total Possible Outcomes
When a single die is thrown, there are 6 possible outcomes (1, 2, 3, 4, 5, or 6). When two dice are thrown simultaneously, the outcomes of each die are independent. Therefore, the total number of possible outcomes is the product of the outcomes for each die.
Total outcomes = Outcomes on Die 1 $\times$ Outcomes on Die 2
Total outcomes = $6 \times 6 = 36$
We can list all the possible outcomes as pairs (outcome on Die 1, outcome on Die 2):
| Die 1 \ Die 2 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
Thus, there are 36 total possible outcomes when two dice are thrown.
Favorable Outcomes: Getting the Same Number
We are looking for the outcomes where the number on the first die is the same as the number on the second die. These are the pairs where both numbers are identical.
The favorable outcomes are:
- (1,1) - getting 1 on both dice
- (2,2) - getting 2 on both dice
- (3,3) - getting 3 on both dice
- (4,4) - getting 4 on both dice
- (5,5) - getting 5 on both dice
- (6,6) - getting 6 on both dice
There are 6 favorable outcomes.
Calculating the Probability
The probability of an event is calculated using the formula:
$$ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} $$
In this case:
Number of favorable outcomes (getting the same number) = 6
Total number of possible outcomes = 36
$$ \text{Probability (getting the same number)} = \frac{6}{36} $$
Now, we simplify the fraction:
$$ \frac{6}{36} = \frac{1 \times 6}{6 \times 6} = \frac{1}{6} $$
So, the probability of getting the same number on both dice is $\frac{1}{6}$.
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