If two balanced dice are tossed once, the probability of the event, that the sum of the integers coming on the upper sides of the two dice is 9, is
1/9
Understanding the Probability of Rolling a Sum of 9 with Two Dice
This question asks for the probability of getting a sum of 9 when tossing two balanced dice once. To solve this, we need to identify the total possible outcomes and the specific outcomes where the sum is 9.
Determining the Sample Space
When a single balanced die is tossed, there are 6 possible outcomes: 1, 2, 3, 4, 5, or 6. When two balanced dice are tossed, the outcome is a pair of numbers, one from each die. The total number of possible outcomes is the product of the number of outcomes for each die.
Total number of outcomes = (Outcomes for Die 1) × (Outcomes for Die 2)
Total number of outcomes = $6 \times 6 = 36$
These 36 outcomes form the sample space. Each outcome is equally likely.
Identifying Favorable Outcomes (Sum is 9)
We are interested in the event where the sum of the integers on the upper sides of the two dice is exactly 9. Let's list the pairs of numbers that add up to 9:
- Die 1 shows 3, Die 2 shows 6 → (3, 6)
- Die 1 shows 4, Die 2 shows 5 → (4, 5)
- Die 1 shows 5, Die 2 shows 4 → (5, 4)
- Die 1 shows 6, Die 2 shows 3 → (6, 3)
These are the only possible combinations where the sum is 9. There are 4 favorable outcomes.
Calculating the Probability
The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability (Sum is 9) = $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$
Probability (Sum is 9) = $\frac{4}{36}$
This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4.
Probability (Sum is 9) = $\frac{4 \div 4}{36 \div 4} = \frac{1}{9}$
So, the probability of the sum being 9 when tossing two balanced dice is 1/9.
Summary of Outcomes
| Sum | Possible Combinations | Number of Outcomes |
|---|---|---|
| 2 | (1, 1) | 1 |
| 3 | (1, 2), (2, 1) | 2 |
| 4 | (1, 3), (2, 2), (3, 1) | 3 |
| 5 | (1, 4), (2, 3), (3, 2), (4, 1) | 4 |
| 6 | (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) | 5 |
| 7 | (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) | 6 |
| 8 | (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) | 5 |
| 9 | (3, 6), (4, 5), (5, 4), (6, 3) | 4 |
| 10 | (4, 6), (5, 5), (6, 4) | 3 |
| 11 | (5, 6), (6, 5) | 2 |
| 12 | (6, 6) | 1 |
| Total | 36 | |
Revision Table: Probability Concepts
| Concept | Description | Formula |
|---|---|---|
| Probability | A measure of the likelihood of an event occurring. | P(Event) = $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ |
| Sample Space | The set of all possible outcomes of an experiment. | Ω or S |
| Event | A specific subset of the sample space that we are interested in. | E <= S |
| Balanced Dice | Dice where each face (1 through 6) has an equal probability of landing face up. | P(any face) = 1/6 |
Additional Information: Dice Rolling Probabilities
Understanding probabilities with dice is a fundamental concept in probability theory. When rolling multiple dice, the total number of outcomes grows exponentially (6n for n dice). The distribution of sums follows a pattern, with sums closer to the middle (like 7 for two dice) having more combinations and thus higher probabilities, while sums at the extremes (like 2 or 12 for two dice) have fewer combinations and lower probabilities.
For two dice, the sum of 7 is the most probable with 6 combinations ((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)), giving a probability of 6/36 = 1/6. The sum of 9, as calculated, has 4 combinations, giving a probability of 4/36 = 1/9.
Important Questions from Probability
The probability of being 53 Sundays in year 2020 is-
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