Question

A coin with P(head) = 2/3 is tossed six times (independently). What is the probability of obtaining exactly four CONSECUTIVE heads or exactly five CONSECUTIVE heads?

Answer 1

Let the obtaining exactly four CONSECUTIVE heads be one event. Then there will be three events, in which one event will result in four consecutive heads.

P(four CONSECUTIVE heads) = (2/3)⁴ = 16/81

The events can be arranged as so as to get exactly four CONSECUTIVE heads

4H, T, H

4H, T, T

T, 4H, T

H, T, 4H

T, T, 4H

Probability of obtaining exactly four CONSECUTIVE heads = 16/81 * 1/3 * 2/3 + 16/81 * 1/3 * 1/3 +   1/3 * 16/81 * 1/3 +  2/3 * 1/3 * 16/81 + 1/3 * 1/3 * 16/81

= 0.1536351

Similarly,

Let the obtaining exactly five CONSECUTIVE heads be one event. Then there will be two events, in which one event will result in five consecutive heads.

P(five CONSECUTIVE heads) = (2/3)⁵ = 32/243

The events can be arranged as so as to get exactly five CONSECUTIVE heads

5H, T

T, 5H

Probability of obtaining exactly five CONSECUTIVE heads = 32/243 * 1/3 + 1/3 * 32/243

= 0.0877915

Since the event of obtaining exactly four CONSECUTIVE heads and obtaining exactly five CONSECUTIVE heads are disjoint events, probability of obtaining exactly four CONSECUTIVE heads or exactly five CONSECUTIVE heads

= 0.1536351 + 0.0877915

= 0.2414266