Calculating the Probability of Obtaining Four or More Heads in 8 Coin Tosses

In many probability and statistics problems, calculating the likelihood of a certain outcome is a common task. One classic example deals with the toss of a fair coin. To find the probability of obtaining four or more heads when a coin is tossed 8 times, we can utilize the binomial probability formula. This article will walk through the detailed steps to calculate this probability and verify it using Excel simulations.

Theoretical Approach

The probability of getting exactly ( k ) heads in ( n ) tosses of a fair coin is given by the binomial probability formula:

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

Where:

( n ) is the total number of trials (coin tosses), ( k ) is the number of successful trials (heads), ( p ) is the probability of success on each trial (for a fair coin ( p 0.5 )), ( binom{n}{k} ) is the binomial coefficient.

In this specific case, with ( n 8 ) and ( p 0.5 ), we want to find the probability of obtaining 4 or more heads. This can be expressed mathematically as:

[ P(X geq 4) P(X 4) P(X 5) P(X 6) P(X 7) P(X 8) ]

Calculating Individual Terms

To find each term, we use the binomial probability formula:

( P(X 4) binom{8}{4} (0.5)^4 (0.5)^4 binom{8}{4} (0.5)^8 frac{70}{256} ) ( P(X 5) binom{8}{5} (0.5)^5 (0.5)^3 binom{8}{5} (0.5)^8 frac{56}{256} ) ( P(X 6) binom{8}{6} (0.5)^6 (0.5)^2 binom{8}{6} (0.5)^8 frac{28}{256} ) ( P(X 7) binom{8}{7} (0.5)^7 (0.5)^1 binom{8}{7} (0.5)^8 frac{8}{256} ) ( P(X 8) binom{8}{8} (0.5)^8 (0.5)^0 binom{8}{8} (0.5)^8 frac{1}{256} )

Summing the Probabilities

Now, we sum up all the individual probabilities:

[begin{align*}P(X geq 4) frac{70}{256} frac{56}{256} frac{28}{256} frac{8}{256} frac{1}{256} frac{70 56 28 8 1}{256} frac{163}{256}end{align*}]

Therefore, the probability of obtaining four or more heads in 8 tosses of a fair coin is:

[ boxed{frac{163}{256}} approx 0.6367 ]

Experimental Approach

To verify this theoretical calculation, we can perform a large number of coin tosses on a computer, simulating the toss of a pseudo-coin in Excel over a million times. The probabilities of obtaining heads can be recorded for each trial. For instance, in our experiment, we obtained the following probabilities:

0 heads: 0.0158 1 head: 0.0946 2 heads: 0.2340 3 heads: 0.3118 4 heads: 0.2345 5 heads: 0.0938 6 heads: 0.0156

The probability of 4 or more heads is the sum of the probabilities for 4, 5, 6, 7, and 8 heads:

[begin{align*}0.2345 0.0938 0.0156 0.34373end{align*}]

This experimental result closely matches the theoretical value of 0.34375, demonstrating the accuracy of our theoretical approach.

Conclusion

In this article, we used both theoretical and experimental approaches to calculate the probability of obtaining four or more heads in 8 coin tosses. We found that the theoretical probability is: