Understanding the Binomial Distribution: Why X=3 is the Most Likely Outcome

Suppose X has a binomial distribution with parameters 6 and 1/2 . Show that X = 3 is the most likely outcome.

The binomial distribution is a fundamental concept in statistics, used to model the number of successes in a fixed number of independent trials with a constant probability of success. In this blog post, we will discuss the binomial distribution with parameters 6 and 1/2, and show why X = 3 is the most likely outcome.

Suppose we have a random variable X that follows a binomial distribution with parameters n = 6 and p = 1/2. This means that X represents the number of successes in six independent trials, where each trial has a 50% chance of success. To understand the likelihood of different outcomes, we can look at the probability density function (PDF) of the binomial distribution:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

where (n choose k) is the binomial coefficient, representing the number of ways to choose k successes out of n trials. Using this formula, we can calculate the probability of each possible outcome for X, ranging from 0 to 6.

X = 0: P(X=0) = (6 choose 0) * (1/2)^0 * (1/2)^6 = 1/64

X = 1: P(X=1) = (6 choose 1) * (1/2)^1 * (1/2)^5 = 6/64

X = 2: P(X=2) = (6 choose 2) * (1/2)^2 * (1/2)^4 = 15/64

X = 3: P(X=3) = (6 choose 3) * (1/2)^3 * (1/2)^3 = 20/64

X = 4: P(X=4) = (6 choose 4) * (1/2)^4 * (1/2)^2 = 15/64

X = 5: P(X=5) = (6 choose 5) * (1/2)^5 * (1/2)^1 = 6/64

X = 6: P(X=6) = (6 choose 6) * (1/2)^6 * (1/2)^0 = 1/64

Notice that the sum of all these probabilities is equal to 1, as expected for any probability distribution.

To find the most likely outcome for X, we can look for the value of k that maximizes the probability P(X = k). This value is known as the mode of the distribution. In this case, we can see that the highest probability occurs for X = 3, with P(X = 3) = 20/64. Therefore, X = 3 is the most likely outcome for the binomial distribution with parameters n = 6 and p = 1/2.

But why is X = 3 the most likely outcome? One way to understand this is by looking at the mean of the distribution, which is given by:

E(X) = n * p

For our binomial distribution with n = 6 and p = 1/2, we have E(X) = 6 * 1/2 = 3. This means that on average, we expect to get 3 successes out of 6 trials. Since the probability distribution is symmetric around the mean, the mode should also occur at the mean, which is the case here with X = 3.