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Omni-MATH / The game of rock-scissors is played just like rock-paper-scissors, except that…

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problem

The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with 50%50 \% probability and scissors with 50%50 \% probability. If you play optimally against the computer, find the probability that after 8 games you have won at least 4.
Plain-text mathematical notation (without MathML)
The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with 50% probability and scissors with 50% probability. If you play optimally against the computer, find the probability that after 8 games you have won at least 4.
Original LaTeX notation
The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with $50 \%$ probability and scissors with $50 \%$ probability. If you play optimally against the computer, find the probability that after 8 games you have won at least 4.

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