If a fair coin is tossed $5$ times, the probability that heads does not occur two or more times in a row is?
- $\displaystyle\frac{12}{2^5}$
- $\displaystyle\frac{13}{2^5}$
- $\displaystyle\frac{14}{2^5}$
- $\displaystyle\frac{15}{2^5}$
The total number of outcomes for 5 coin tosses is 2^5 = 32. Using recursion or counting, the number of sequences without 'HH' is the (n+2)-th Fibonacci number, which for n=5 is F(7) = 13.
The total number of outcomes when a fair coin is tossed 5 times is 2^5 = 32. To find the number of favorable outcomes where heads does not occur twice in a row, we can systematically list the valid sequences based on their number of tails: 5 tails (TTTTT) yields 1 sequence, 4 tails yield 5 sequences, and 3 tails yield 10 sequences. Since any valid sequence must contain at least 3 tails to separate the heads, the total number of favorable sequences is 1 + 5 + 10 = 13. Therefore, the required probability is 13 divided by 32, which equals 13 out of 2^5.