An unbiased coin is tossed 8 times. The probability that head appears consecutively at least 5 times is
Show Hint
- \( \frac{5}{256} \)
- \( \frac{5}{128} \)
- \( \frac{5}{64} \)
- \( \frac{5}{32} \)
The Correct Option is B
Solution and Explanation
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 - \( \text{HHHHHHH.
.
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} \), 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).
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