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What is the probability of getting six heads when you toss a coin ten times? \newlineWrite your answer as a percentage, and round to the nearest hundredth.\newline

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Q. What is the probability of getting six heads when you toss a coin ten times? \newlineWrite your answer as a percentage, and round to the nearest hundredth.\newline
  1. Understand the problem: Understand the problem.\newlineWe need to calculate the probability of getting exactly six heads when a fair coin is tossed ten times. This is a binomial probability problem, where the number of trials is 1010, the number of successes (heads) we want is 66, and the probability of success on a single trial (getting a head on a single coin toss) is 0.50.5.
  2. Use the binomial probability formula: Use the binomial probability formula.\newlineThe binomial probability formula is P(X=k)=(nk)⋅pk⋅(1−p)n−kP(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}, where P(X=k)P(X = k) is the probability of kk successes in nn trials, (nk)\binom{n}{k} is the binomial coefficient, pp is the probability of success on a single trial, and (1−p)(1-p) is the probability of failure on a single trial.
  3. Calculate the binomial coefficient: Calculate the binomial coefficient ((106))(10 \choose 6).\newlineThe binomial coefficient ((nk))(n \choose k) is calculated as n!k!⋅(n−k)!\frac{n!}{k! \cdot (n-k)!}, where n!n! is the factorial of nn.\newline((106))(10 \choose 6) = 10!6!⋅(10−6)!\frac{10!}{6! \cdot (10-6)!} = 10!6!⋅4!\frac{10!}{6! \cdot 4!} = 10⋅9⋅8⋅74⋅3⋅2⋅1\frac{10\cdot9\cdot8\cdot7}{4\cdot3\cdot2\cdot1} = 210210.
  4. Calculate the probability of getting exactly six heads: Calculate the probability of getting exactly six heads.\newlineUsing the binomial probability formula:\newlineP(X=6)=(106)×(0.5)6×(0.5)10−6P(X = 6) = \binom{10}{6} \times (0.5)^6 \times (0.5)^{10-6}\newlineP(X=6)=210×(0.5)6×(0.5)4P(X = 6) = 210 \times (0.5)^6 \times (0.5)^4\newlineP(X=6)=210×(0.5)10P(X = 6) = 210 \times (0.5)^{10}\newlineP(X=6)=210×(1/1024)P(X = 6) = 210 \times (1/1024)\newlineP(X=6)=210/1024P(X = 6) = 210 / 1024\newlineP(X=6)=0.205078125P(X = 6) = 0.205078125
  5. Convert the probability to a percentage and round: Convert the probability to a percentage and round to the nearest hundredth.\newlineTo convert to a percentage, we multiply by 100100.\newlinePercentage = 0.205078125×1000.205078125 \times 100\newlinePercentage ≈20.51%\approx 20.51\%

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