A Coin Is Tossed 72 Times Find The Standard Deviation For It

A Coin Is Tossed 72 Times Find The Standard Deviation For The Number

A coin is tossed 72 times. Find the standard deviation for the number of heads that will be tossed.

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When a fair coin is tossed multiple times, the number of heads obtained can be modeled using a binomial distribution. Understanding this distribution allows us to calculate the variability of the outcomes, specifically through the standard deviation, which measures how much the number of heads is expected to fluctuate around the mean.

In this scenario, each coin toss is a Bernoulli trial, with only two possible outcomes: heads or tails. The probability of getting heads in a single toss (p) is 0.5 for a fair coin. When conducting 72 independent tosses, the total number of heads follows a binomial distribution characterized by parameters n = 72 and p = 0.5.

The mean (expected value) of the binomial distribution is calculated as:

Mean (μ) = n p = 72 0.5 = 36

The standard deviation (σ) of a binomial distribution is given by the formula:

σ = sqrt(n p (1 - p))

Substituting the known values:

σ = sqrt(72 0.5 0.5) = sqrt(72 * 0.25) = sqrt(18) ≈ 4.24

Therefore, the standard deviation of the number of heads in 72 coin tosses is approximately 4.24.

This measure indicates the typical deviation from the average of 36 heads, providing insight into the variability of the outcomes in repeated sequences of 72 coin flips. When interpreting the results, if the actual number of heads reported in a particular set of 72 tosses deviates significantly from 36, understanding the standard deviation helps determine whether such deviations are likely or indicative of a bias or problem with the fairness of the coin.

In conclusion, the use of the binomial model is fundamental in probability and statistics for analyzing binary outcomes. Knowledge of the standard deviation allows statisticians and practitioners to assess the expected range of results and make informed judgments regarding the fairness, randomness, or particular trends in experimental or real-world processes involving binary events.

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