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If a fair coin is tossed times, what is the probability, rounded to the nearest thousandth, of getting at least tails?Answer:
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Q. If a fair coin is tossed times, what is the probability, rounded to the nearest thousandth, of getting at least tails?Answer:
- Calculate Probability of Tails: Determine the probability of getting exactly tails in tosses.The probability of getting a tail in one toss is . Since the coin is fair, each toss is independent of the others. To get exactly tails, one of the tosses must be a head. There are different positions where the head can occur (first toss, second toss, ..., ninth toss). The probability of getting exactly tails and head in a specific order is $(\frac{\(1\)}{\(2\)})^\(8\) \times (\frac{\(1\)}{\(2\)})^\(1\) = (\frac{\(1\)}{\(2\)})^\(9\).
- Calculate Probability of \(8\) Tails: Calculate the probability of getting exactly \(8\) tails. Since there are \(9\) different positions for the single head, we multiply the probability of getting \(8\) tails and \(1\) head in a specific order by \(9\). Probability of exactly \(8\) tails = \(9 \times (\frac{1}{2})^9\)
- Calculate Probability of \(9\) Tails: Determine the probability of getting exactly \(9\) tails in \(9\) tosses.\(\newline\)The probability of getting a tail on each toss is \((\frac{1}{2})^9\), since all \(9\) tosses must be tails.
- Calculate Total Probability: Calculate the total probability of getting at least \(8\) tails. This is the sum of the probabilities of getting exactly \(8\) tails and exactly \(9\) tails. Total probability = Probability of exactly \(8\) tails + Probability of exactly \(9\) tails Total probability = \(9 \times (\frac{1}{2})^{9} + (\frac{1}{2})^{9}\)
- Perform Calculation: Perform the calculation.\(\newline\)Total probability = \(9 \times \left(\frac{1}{2}\right)^9 + \left(\frac{1}{2}\right)^9\)\(\newline\)Total probability = \(\frac{9}{512} + \frac{1}{512}\)\(\newline\)Total probability = \(\frac{10}{512}\)\(\newline\)Total probability = \(0.01953125\)
- Round Total Probability: Round the total probability to the nearest thousandth.\(\newline\)Rounded probability = \(0.020\)
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