Solving the Probability of a Head Appearing on the Fifth Trial of a Fair Coin Flip

Probability is a branch of mathematics that deals with the likelihood of an event occurring. In the case of coin flipping, it can help us understand the chances of getting a specific outcome. In this blog post, we will solve the probability of a head appearing on the fifth trial of a fair coin flip.

If a fair coin is successively flipped, find the probability that a head first appears on the fifth trial.

The probability of getting a head or a tail on a single coin flip is 1/2 or 0.5. Since the coin is fair, the probability of getting a head or a tail remains the same for every flip.

To find the probability of getting a head on the fifth trial, we need to consider all possible outcomes for the first four flips. There are two possible outcomes for each flip, so there are 2^4 = 16 possible sequences of four flips.

Out of these 16 sequences, there is only one sequence that ends with a head on the fifth trial: TTTTH, where T represents a tail and H represents a head. Therefore, the probability of getting a head on the fifth trial is 1/16 or 0.0625.

We can also use the formula for the geometric probability distribution to find the probability of getting a head on the fifth trial. The formula is:

P(X=k) = (1-p)^(k-1) * p

Where X is the random variable representing the number of trials until the first success (getting a head in this case), k is the specific trial we are interested in (fifth trial in this case), p is the probability of success (getting a head in this case), and (1-p) is the probability of failure (getting a tail in this case).

Using this formula, we get:

P(X=5) = (1-0.5)^(5-1) * 0.5 = 0.0625

which confirms our previous result.

The probability of getting a head on the fifth trial of a fair coin flip is 1/16 or 0.0625. This result is important because it helps us understand the likelihood of getting a specific outcome when flipping a fair coin. Knowing the probability of an event can be useful in many areas, such as gambling, insurance, and scientific research.

Overall, probability theory is an essential tool in understanding the world around us and can be applied in many fields. By understanding how to calculate probabilities, we can make more informed decisions and predictions.

References:

Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Boston, MA: Cengage Learning.

Ross, S. M. (2010). A First Course in Probability (8th ed.). Upper Saddle River, NJ: Pearson Education.