Probability of getting heads twice in a row vs. The same number twice when rolling a dice

Question

Can someone explain to me that when calculating the odds of flipping a coin twice and it landing heads both times, the formula is $\frac 12 \cdot \frac 12$ or $0.5 \times 0.5$ for a result of $0.25$ but when calculating the odds of rolling a dice and it landing on the same number twice it's not $\frac 16 \cdot \frac 16$ or $0.166 \times 0.166$ for a result of $0.027$? I know the answer for the dice is $1/6$ but I'm just confused as to why you can't use the same formula to figure out both.

Answer 1

What's important is how many (equally likely) events there are and what events fulfill the desired outcome. For tossing a coin 2 times there are 4 possible outcomes (respecting the order) and only one event is favourable (heads, heads). Hence the probability is $1/4$. For the dice, there are $6 \times 6 = 36$ possible outcomes (respecting the order) and 6 of them fulfil the desired result (that's $(1, 1), (2, 2), \ldots, (6, 6)$). Hence the probability is $6/36 = 1/6$.

Answer 2

Let's analyze why the first probability is $0.5\times 0.5$:
What is the probability that you get a head on the first coin? That's $\color{blue}{0.5}$. What is the probability that you get a head on the second coin? Again, it's $\color{blue}{0.5}$, so multiply both to get the final probability.

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Let's analyze why the second probability is $1/6$:
Using the same logic, what is the probability you get a -- number on the dice? Of course, it's $\color{green}{1}$, because you are sure to get some number. Let's call whatever number appeared on the first dice as $x$. What is the probability that $x$ appears on the second dice? It's $\color{green}{1/6}$. Now, multiply both of them to get the required probability.

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If you don't get it from the boldfaced text: The difference is that for the dice case, you don't have any fixed number. For the coin case, you were given that you want heads on both coins. But in the dice case, you want any number on the first dice, and the first number on the second dice.
Say you were given the question:

You roll $2$ dice, what is the probability that you get $\mathbf 5$ on both the dice?

Here, the answer should be $\frac 16 \times \frac 16$.