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35 votes
If four dice are rolled, what is the probability of obtaining two identical odd numbers and two identical even numbers?

User Skwashua
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1 Answer

12 votes
12 votes

Answer:

1/24 = 0.041666666...

Explanation:

when rolling 4 dice, we have

6 × 6 × 6 × 6 = 6⁴ = 1296

different possible outcomes (each die has 6 possible outcomes).

so, the probability of any particular pattern is then

1/1296.

we need to find how many patterns there are to represent the desired result of 2 identical odd numbers combined with 2 identical even numbers.

there are on every die 3 out of 6 options for an odd number, and the other 3 out of 6 options for an even number.

when we roll 2 dice, the patterns for 2 identical odd numbers are

1 1

3 3

5 5

so, 3 out of the possible 36.

similar for 2 identical even numbers.

2 2

4 4

6 6

3 out of the possible 36.

so, combining this together for a roll of 4 dice, we get then

3 × 3 out of the possible 36 × 36

9 out of the possible 1296 patterns.

that means the probability (desired over total possible cases) for a specific combination of 2-and-2 dice is

9/1296 = 1/144 = 0.006944444...

but there are 4! / (2! × 2!) = 6 possibilities (pick 2 out of 4, where the sequence of the picked ones does not matter) to pick 2 dice out of 4 and to show the same pattern across the 4 dice, as e.g. for the basic pattern 1 1 2 2 we can have

1 1 2 2

1 2 1 2

1 2 2 1

2 1 2 1

2 1 1 2

2 2 1 1

all of the 9 patterns have these 6 combination possibilities.

so, we need to multiply the individual pattern probability by 6 for the total probability

1/144 × 6 = 6/144 = 1/24 = 0.041666666...

User Catwell
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