Questions: What is the probability of obtaining seven heads in a row when flipping a coin? Interpret this probability. The probability of obtaining seven heads in a row when flipping a coin is . (Round to five decimal places as needed.)

What is the probability of obtaining seven heads in a row when flipping a coin? Interpret this probability.

The probability of obtaining seven heads in a row when flipping a coin is .
(Round to five decimal places as needed.)

Transcript text: What is the probability of obtaining seven heads in a row when flipping a coin? Interpret this probability. The probability of obtaining seven heads in a row when flipping a coin is $\square$ . (Round to five decimal places as needed.)

Solution

Solution Steps

Step 1: Define the Problem

We want to find the probability of obtaining seven heads in a row when flipping a fair coin. This can be modeled using the binomial distribution, where the number of trials $ n = 7 $ and the number of successes $ x = 7 $.

Step 2: Set the Parameters

For a fair coin:

The probability of success (getting heads) is $ p = 0.5 $.
The probability of failure (getting tails) is $ q = 1 - p = 0.5 $.

Step 3: Apply the Binomial Probability Formula

The probability of obtaining exactly $ x $ successes in $ n $ trials is given by the formula: \[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \] Substituting the values: \[ P(X = 7) = \binom{7}{7} \cdot (0.5)^7 \cdot (0.5)^{0} = 1 \cdot (0.5)^7 \cdot 1 = (0.5)^7 \]

Step 4: Calculate the Probability

Calculating $ (0.5)^7 $: \[ (0.5)^7 = \frac{1}{128} \approx 0.0078125 \] Rounding to five decimal places, we find: \[ P(X = 7) \approx 0.00781 \]

Step 5: Interpret the Result

The probability of obtaining seven heads in a row when flipping a fair coin is approximately $ 0.00781 $. This indicates that such an event is quite rare, occurring roughly 0.781% of the time.