Six coins are tossed 6,400 times. Using the Poisson distribution, find the approximateprobability of getting six heads x times.

To find the approximate probability of getting six heads x times using the Poisson distribution, we can follow these steps:

1. Determine the average number of successes (heads) per trial. In this case, since there are six coins and each coin has two possible outcomes (heads or tails), the average number of heads per trial is 6/2 = 3.

2. Calculate the parameter λ (lambda) for the Poisson distribution. Lambda is equal to the average number of successes per trial. In this case, λ = 3.

3. Use the Poisson probability formula to calculate the probability of getting exactly x successes (six heads) in a given number of trials (6,400 tosses). The formula is:

   P(x) = (e^(-λ) * λ^x) / x!

   where e is the base of the natural logarithm (approximately 2.71828), λ is the parameter, x is the number of successes, and x! is the factorial of x.

4. Substitute the values of λ and x into the formula and calculate the probability for each value of x.

   For example, to find the probability of getting exactly six heads (x = 6) in 6,400 tosses, we have:

   P(6) = (e^(-3) * 3^6) / 6!

5. Repeat step 4 for different values of x to find the probabilities for each possible outcome.

Note: The Poisson distribution is an approximation for the binomial distribution when the number of trials is large and the probability of success is small. In this case, the number of trials (6,400) is large, and the probability of getting a head on each coin toss is 1/2, which is considered small.