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# Difficult Counting Problem

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Gracie starts at her house and walkes one block east. After each block she walks, she flips a coin. She walks one block east if the coin lands heads; otherwise one block west. She stops her walk the first time she lands back home. What is the probability she stops her walk after walking exactly eight blocks? Express your answer as a common fraction.

P.S It would be very helpful if you could include a in-depth solution

Aug 13, 2024

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Analyzing the Problem

To return home after exactly 8 blocks, Gracie must walk 4 blocks east and 4 blocks west. The order of these blocks doesn't matter as long as she ends up at home.

Calculating the Probability

Total possible outcomes: There are 2 possible outcomes (heads or tails) for each of the 8 blocks, so there are 2^8 = 256 total possible outcomes.

Favorable outcomes: We need to choose 4 out of the 8 blocks to be eastbound (the remaining 4 will be westbound). This is a combination problem. The number of ways to choose 4 blocks out of 8 is 8C4 = 70.

Calculating the Probability

Probability = Favorable outcomes / Total possible outcomes

Probability = 70 / 256 = 35/128

Therefore, the probability that Gracie stops her walk after walking exactly eight blocks is 35/128.

Aug 13, 2024