A fair coin is tossed $12$ times. Find the probability that two heads do not occur consecutively.
- Required probability is $\cfrac{375}{4096}$
- Required probability is $\cfrac{376}{4096}$
- Required probability is $\cfrac{377}{4096}$
- Required probability is $\cfrac{383}{4096}$
Reveal answer
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Correct answer
Explanation
The number of ways to toss a coin 12 times such that no two heads are consecutive is given by the (n+2)-th Fibonacci number, where n=12. F(14) = 377. Total outcomes = 2^12 = 4096. Probability = 377/4096.
AI explanation
The total number of possible outcomes when a fair coin is tossed 12 times is 2^12, which equals 4096. The number of outcomes where two heads do not occur consecutively corresponds to the sum of binomial coefficients for selecting up to 6 non-consecutive heads, which evaluates to 12C0 + 12C1 + 11C2 + 10C3 + 9C4 + 8C5 + 7C6, equaling 377. The required probability is the number of favorable outcomes divided by the total outcomes, which is 377/4096.