Math  /  Data & Statistics

Question66 A pair of fair and ordinary dice is rolled simultaneously. It is found that they show different outcomes. The probability that the sum of the outcomes will be either 4 or 6 or 8 is equal to

Studdy Solution

STEP 1

1. Each die has 6 faces numbered from 1 to 6.
2. The dice are fair, meaning each face has an equal probability of landing face up.
3. The outcomes of the dice are different.
4. We are interested in the probability that the sum of the outcomes is 4, 6, or 8.

STEP 2

1. Calculate the total number of possible outcomes where the dice show different numbers.
2. Identify the successful outcomes where the sum is 4, 6, or 8.
3. Calculate the probability by dividing the number of successful outcomes by the total number of possible outcomes.

STEP 3

Calculate the total number of possible outcomes where the dice show different numbers.
Each die has 6 faces, so there are 6×6=366 \times 6 = 36 total possible outcomes when rolling two dice. However, we need to exclude outcomes where both dice show the same number (e.g., (1,1), (2,2), ..., (6,6)). There are 6 such outcomes.
Total different outcomes:
366=30 36 - 6 = 30

STEP 4

Identify the successful outcomes where the sum is 4, 6, or 8.
- Sum of 4: Possible pairs are (1,3), (3,1), (2,2) (but (2,2) is not allowed as the numbers must be different), so successful pairs are (1,3), (3,1). - Sum of 6: Possible pairs are (1,5), (5,1), (2,4), (4,2), (3,3) (but (3,3) is not allowed), so successful pairs are (1,5), (5,1), (2,4), (4,2). - Sum of 8: Possible pairs are (2,6), (6,2), (3,5), (5,3), (4,4) (but (4,4) is not allowed), so successful pairs are (2,6), (6,2), (3,5), (5,3).
Total successful outcomes:
2+4+4=10 2 + 4 + 4 = 10

STEP 5

Calculate the probability by dividing the number of successful outcomes by the total number of possible outcomes.
Probability:
1030=13 \frac{10}{30} = \frac{1}{3}
The probability that the sum of the outcomes will be either 4, 6, or 8 is:
13 \boxed{\frac{1}{3}}

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