Question

Two dice are rolled together. The probability of getting an outcome (x, y) where x > y, is ______.

पर्याय

• `5/12`

• `5/6`

• 1

• 0

Answer 1

Two dice are rolled together. The probability of getting an outcome (x, y) where x > y, is `underlinebb(5/12)`.

Explanation:

1. Identify the total outcomes

When two dice are rolled, each die has 6 possible results. The total number of outcomes in the sample space S is calculated by multiplying the possibilities of each die:

Total outcomes = 6 × 6 = 36

2. Categorize the possibilities

The 36 possible outcomes (x, y) can be split into three distinct groups:

1. x = y: Cases where both dice show the same number. There are 6 such outcomes: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6).
2. x ≠ y: The remaining 36 – 6 = 30 outcomes.
3. Symmetry: Within the x ≠ y group, half the outcomes will satisfy x > y and the other half will satisfy x < y because the dice are identical in their distribution.

3. Calculate the favorable outcomes

To find the number of outcomes where x > y, we take the total non-equal outcomes and divide by 2: 

Favaorable outcomes = `30/2 = 5`

4. Determine the probability

The probability P is the ratio of favorable outcomes to the total outcomes:

`P(x > y) = 15/36`

Simplifying the fraction by dividing both the numerator and the denominator by 3 gives:

`P(x > y) = 5/12`