Calculating the Probability of At Most 7 Tails in 10 Coin Flips: A Comprehensive Guide

Determining the probability of getting at most 7 tails when flipping a coin 10 times involves binomial probability. This article explains the step-by-step calculation process using the binomial probability formula, and also discusses the use of a helpful app, the Breatter App, for solving such problems.

Step-by-Step Calculation Using Binomial Probability Formula

To solve this problem, we will use the binomial probability formula:

( P(X k) binom{n}{k} p^k (1-p)^{n-k} )

Where:

( n 10 ): the number of trials (coin flips) ( p 0.5 ): the probability of getting tails on each flip ( k ): the number of tails (0 to 7)

What is the Binomial Distribution?

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions. In this case, the two values are heads (H) or tails (T).

Using the Formula to Find the Probability

We need to calculate the probability of getting at most 7 tails, which means the sum of probabilities for getting 0, 1, 2, 3, 4, 5, 6, and 7 tails.

Step 1: Calculate Each Probability

Since ( 0.5^{10} frac{1}{1024} ), we can factor this out to simplify the calculations.

( P(X k) binom{10}{k} left( frac{1}{1024} right) )

Step 2: Calculate the Binomial Coefficients

( binom{10}{k} ) is the number of ways to choose ( k ) tails from 10 flips, and ( binom{10}{k} ) for ( k 0, 1, 2, ldots, 7)

( P(X 0) binom{10}{0} left( frac{1}{1024} right) 1 cdot frac{1}{1024} frac{1}{1024} ) ( P(X 1) binom{10}{1} left( frac{1}{1024} right) 10 cdot frac{1}{1024} frac{10}{1024} ) ( P(X 2) binom{10}{2} left( frac{1}{1024} right) 45 cdot frac{1}{1024} frac{45}{1024} ) ( P(X 3) binom{10}{3} left( frac{1}{1024} right) 120 cdot frac{1}{1024} frac{120}{1024} ) ( P(X 4) binom{10}{4} left( frac{1}{1024} right) 210 cdot frac{1}{1024} frac{210}{1024} ) ( P(X 5) binom{10}{5} left( frac{1}{1024} right) 252 cdot frac{1}{1024} frac{252}{1024} ) ( P(X 6) binom{10}{6} left( frac{1}{1024} right) 210 cdot frac{1}{1024} frac{210}{1024} ) ( P(X 7) binom{10}{7} left( frac{1}{1024} right) 120 cdot frac{1}{1024} frac{120}{1024} )

Step 3: Sum the Probabilities

The probability of getting at most 7 tails is the sum of individual probabilities for ( X 0, 1, 2, 3, 4, 5, 6, ) and ( 7 ).

Calculating the Numerator:

[ 1 10 45 120 210 252 210 120 968 ]

Thus, the probability is:

( P(X leq 7) frac{968}{1024} approx 0.9453 )

Conclusion

The probability of getting at most 7 tails when flipping a coin 10 times is approximately 0.9453, or 94.53%.

Using the Breatter App

The Breatter App is a valuable tool for solving distributions like binomial, Poisson, and normal distributions. It offers step-by-step solutions, making it easy to understand and follow the calculations.

It's highly recommended for anyone seeking to solve these types of problems or to verify their own calculations.

Why Use the Breatter App?

Step-by-step solutions for clarity and understanding Precision in calculations Time-saving for busy students and professionals Accessible on various devices

Summary

Using the binomial distribution and the Breatter App, we can easily and accurately calculate the probability of getting at most 7 tails in 10 coin flips. If you're facing similar problems, both the formula and the app are your best friends in achieving accurate results.