Solution: We are to count the number of binary strings of length 5 (using 'E' and 'J', or 0 and 1), where no two adjacent characters are both 'J'. This is equivalent to counting binary strings without consecutive 1s.
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- \frac{63}{384} = \frac{21}{128}.
- Thus, the probability that the number of heads equals the die roll is $ \boxed{\dfrac{21}{128}} $.
- Question: An elementary school student is learning about the solar system and draws 5 planets, coloring each one either Earth-like (E) or Jupiter-like (J), but insists that no two adjacent planets in her linear diagram are both Jupiter-like. How many valid colorings are possible?
- Let $ a_n $ be the number of valid colorings of $ n $ planets with no two adjacent Jupiter-like planets.
- This satisfies the Fibonacci recurrence:
- a_n = a_{n-1} + a_{n-2},
