The rolls of the dice are independent, i.e. the outcome of the second die doesn't depend in any way on the outcome of the first die.
In cases like this, the probability of two events happening one after the other is the multiplication of the probabilities of the two events.
So, the probability of rolling two 6s is the multiplication of the probabilities of rolling a six with the first die, and another six with the second:
[tex] P(\text{rolling two 6s}) = P(\text{rolling a 6}) \cdot P(\text{rolling a 6}) = \dfrac{1}{6} \cdot \dfrac{1}{6} = \dfrac{1}{36} [/tex]
Similarly,
[tex] P(\text{rolling two 3s}) = P(\text{rolling a 3}) \cdot P(\text{rolling a 3}) = \dfrac{1}{6} \cdot \dfrac{1}{6} = \dfrac{1}{36} [/tex]
Actually, you can see that the probability of rolling any ordered couple is always 1/36, since the probability of rolling any number on both dice is 1/6:
[tex] P(\text{rolling any ordered couple}) = P(\text{rolling the first number}) \cdot P(\text{rolling the second number}) = \dfrac{1}{6} \cdot \dfrac{1}{6} = \dfrac{1}{36} [/tex]
The probabilities of obtaining 6s and 3s from the roll of the fair dice are same, thus the likelihood of both outcomes are the equal.
The roll of 3s and 6s are equally likely :
To demonstrate this, we calculate the probability of rolling 6s and 3s on 2 rolls of a dice :
The sample space for a single roll of 2 fair dice is attached below :
Total sample space = 36
Recall :
Probability = required outcome / Total possible outcomes
Probability of obtaining 6s from a single roll of 2 dice :
P(6's) = n(6,6) / sample space = 1 / 36
Therefore, probability of obtaining 6s from 2 rolls of the dice is :
First roll × second roll
p(6's) × p(6's) = 1/36 × 1/36 = 1 / 1296
Similarly :
Probability of obtaining 3's from a single roll of 2 dice :
P(3's) = n(3,3) / sample space = 1 / 36
Therefore, probability of obtaining 3's from 2 rolls of the dice is :
First roll × second roll
p(3's) × p(3's) = 1/36 × 1/36 = 1 / 1296
Therefore, since the outcome of each event have equal probabilities, then the rolls of 6's and 3's are equally likely .
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