Problem 51 - College Algebra

Question

If you toss a fair coin six times, what is the probability of getting all heads?

Step-by-Step Solution

Answer

The probability of getting all heads when a fair coin is tossed six times is \(\frac{1}{64}\)

Step 1 - Identify Outcomes for Single Event

For a single coin toss, there are two possible outcomes: heads or tails. Since it is a fair coin, the probability of getting a head (H) is \(\frac{1}{2}\) and similarly, the probability of getting a tails (T) is also \(\frac{1}{2}\). This is an example of a biased coin. If it were an unbiased coin, the probabilities for head and tails would not be equal.

Step 2 - Calculate for Multiple Independent Events

Now, we want to know the probability of getting six heads in a row. Since each toss is an independent event, meaning the result of one toss does not affect the others, we can find this by multiplying the probability of getting a head in a single toss six times. This is \(\left(\frac{1}{2}\right) * \left(\frac{1}{2}\right) * \left(\frac{1}{2}\right) * \left(\frac{1}{2}\right) * \left(\frac{1}{2}\right) * \left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^6\).

Step 3 - Simplify the Expression

Next, simplify the product \(\left(\frac{1}{2}\right)^6 = \frac{1}{64}\). So, the probability of getting all heads when you toss a fair coin six times is \(\frac{1}{64}\).