If you flip six unbiased coins simultaneously, what is the likelihood of exactly three of them landing on heads?

To solve this problem, we need to use the concept of binomial probability. The formula for binomial probability is:

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

where:

• P(X=k) is the probability of k successes in n trials
• C(n, k) is the combination of n items taken k at a time
• p is the probability of success on a single trial
• n is the number of trials
• k is the number of successes

In this case, we are flipping 6 coins (n=6), and we want to find the probability of getting exactly 3 heads (k=3). The probability of getting a head on a single coin flip is 0.5 (p=0.5), because a coin has 2 equally likely outcomes.

So, we can plug these values into the formula:

P(X=3) = C(6, 3) * (0.5^3) * ((1-0.5)^(6-3))

First, calculate C(6, 3). This is the number of ways to choose 3 items from 6, which is 6! / [3!(6-3)!] = 20.

Next, calculate (0.5^3) = 0.125.

Then, calculate ((1-0.5)^(6-3)) = (0.5^3) = 0.125.

Finally, multiply these values together:

P(X=3) = 20 * 0.125 * 0.125 = 0.3125.

So, the probability of getting exactly 3 heads when flipping 6 unbiased coins is 0.3125, or 31.25%.