If Benjie throws a coin until a series of three consecutive heads or three consecutive tails appears, what is the probability that the game will end on the fourth throw?

# In tossing a coin, what is the probability of getting three consecutive heads or tails

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We can use the geometric distribution to get the probability of an event (success) occurring after a number of failures.

If repeated independent trials can result in a success with probability $p$ and a failure with probability $q = 1 - p$, then the probability distribution of the random variable $X$, the number of trials on which the first success occurs, is:

$$g(x;p) = pq^{x-1}$$

and $x = 1,2,3,4,5,.....$

In this context, the first success is getting three consecutive heads or three consecutive tails in flipping coins. The probability of getting a head (or tail) in coin toss is $p = 0.5$. The probability that the first success occurs after flipping the coin four times would be:

$$g(x;p) = pq^{x-1}$$ $$g(4;0.5) = 0.5(1-0.5)^{(4)-1} = 0.0625 $$

Therefore, the probability that the game will end on the fourth throw is $\color{green}{6.25 \%}$

Alternate solutions are encouraged....