Question

An unbiased coin is tossed 8 times. The probability that head appears consecutively at least 5 times is

• A. \( \frac{5}{256} \)
• B. \( \frac{5}{128} \)
• C. \( \frac{5}{64} \)
• D. \( \frac{5}{32} \)

Hint:

For "at least k consecutive heads" problems, list all disjoint patterns of runs of exactly \( k \) heads, and sum the cases. The total favorable outcomes divided by total possible outcomes gives the probability.

Answer 1

The Correct Option is B

The total number of outcomes for 8 tosses is \(2^8 = 256\).
We need the probability of at least 5 consecutive heads.
Consider cases for the number of consecutive heads: - Case 1: Exactly 5 heads - \( \text{HHHHHT.
.
.
} \), where the remaining tosses can be either heads or tails: \( 2^2 = 4 \) ways.
- Case 2: Exactly 6 heads - \( \text{HHHHHHT.
.
.
} \), leading to 3 ways.
- Case 3: Exactly 7 heads - \( \text{HHHHHHH.
.
.
} \), leading to 2 ways.
- Case 4: Exactly 8 heads - \( \text{HHHHHHHH} \), 1 way.
Thus, the total number of favorable outcomes is \( 1 + 2 + 3 + 4 = 10 \).
The probability is \( \frac{10}{256} = \frac{5}{128} \), which matches option (2).