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?

April 19, 2017 - 7:26am

#1
In tossing a coin, what is the probability of getting three consecutive heads or tails

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?

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....

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