Probability of Eight Heads in a Row: 0.00391 and Interpretation

What is the probability of obtaining eight heads in a row when flipping a​ coin? Interpret this probability. Question content area bottom Part 1 The probability of obtaining eight heads in a row when flipping a coin is    enter your response here. ​(Round to five decimal places as​ needed.) Part 2 Interpret this probability. Consider the event of a coin being flipped eight times. If that event is repeated ten thousand different​ times, it is expected that the event would result in eight heads about    enter your response here ​time(s). ​(Round to the nearest whole number as​ needed.)

To calculate the probability of obtaining eight heads in a row when flipping a fair coin, we can use the following reasoning:

1. The probability of getting heads on a single flip of a fair coin is ( \frac{1}{2} ).
2. Since the flips are independent, the probability of getting heads eight times in a row is:

[ P(\text{8 heads}) = \left(\frac{1}{2}\right)^8 = \frac{1}{256} ]

Now, let's calculate this value:

[ \frac{1}{256} \approx 0.00390625 ]

Rounding to five decimal places, we get:

[ P(\text{8 heads}) \approx 0.00391 ]

Part 1

The probability of obtaining eight heads in a row when flipping a coin is 0.00391.

Part 2

To interpret this probability, we consider the event of flipping a coin eight times. If this event is repeated 10,000 different times, we can expect the number of times we get eight heads in a row to be:

[ \text{Expected occurrences} = P(\text{8 heads}) \times \text{Number of trials} = 0.00390625 \times 10000 ]

Calculating this gives:

[ 0.00390625 \times 10000 = 39.0625 ]

Rounding to the nearest whole number, we get:

[ \text{Expected occurrences} \approx 39 ]

Part 2 Answer

If that event is repeated ten thousand different times, it is expected that the event would result in eight heads about 39 time(s).