Final answer:
The long-term probability of getting exactly 3 heads in 5 flips of a fair coin is 31.25%. This is calculated using the binomial distribution formula, which takes into account the number of possible outcomes, the probability of a single outcome, and the number of ways to achieve exactly 3 heads.
Step-by-step explanation:
Long-Term Probability of Exactly 3 Heads in 5 Coin Flips
When a fair coin is flipped, the theoretical probability of getting a head (or a tail) on any single flip is 0.5, regardless of previous outcomes. This concept is based on the independence of flips in probability theory. Understanding the law of large numbers, as demonstrated by Karl Pearson, reassures that in the long term, the relative frequency of an event will approach its theoretical probability.
To calculate the probability of getting exactly 3 heads in 5 flips of a fair coin, we use the binomial distribution formula:
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- Identify the number of possible outcomes (n=5)
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- Determine the probability of getting a head on a single coin toss (p=0.5)
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- Calculate the number of ways 3 heads can occur among 5 flips
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- Compute the probability using the binomial formula: P(X=k) = nCk * (pk) * (1-p)(n-k)
For exactly 3 heads, the computation is 5C3 * (0.5)3 *(0.5)5-3 which equals 10 * 0.125 * 0.25 = 0.3125 or 31.25%. Therefore, the long-term probability of obtaining exactly 3 heads in 5 flips of a fair coin is 31.25%.
In conclusion, using the principles of theoretical probability and binomial distribution, one can calculate the long-term likelihood of specific outcomes from repetitive independent trials like coin flips.