Problem 50 - College Algebra

Question

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

Step-by-Step Solution

Answer

The probability of getting all tails when tossing a fair coin seven times is \(\frac{1}{128}\).

Step 1: Identify the probability for a single event

A fair coin has two equally likely outcomes: heads or tails. Therefore, the probability of getting tails (T) in one coin toss is \( P(T) = \frac{1}{2} \).

Step 2: Multiply the probabilities for independent events

Given that the coin tosses are independent, the probability of getting all tails in seven tosses is the product of the probabilities of getting tails in each toss. It can be computed using the formula \( P(A \cap B) = P(A)P(B) \) for independent events A and B, extrapolated to our 7 identical, independent events. So \( P(T_7) = P(T)^7 = \left(\frac{1}{2}\right)^7 \).

Step 3: Compute the final result

We calculate the expression found above, so our final result is \( P(T_7) = \left(\frac{1}{2}\right)^7 = \frac{1}{128} \).