What is the probability of flipping exactly 3 heads in 5 coin flips?

Official Answer

Certainly, as a Data Scientist, I approach probability questions by breaking down the problem into smaller, more manageable components, applying foundational principles of probability and combinatorics to solve them. Let's consider the question at hand: What is the probability of flipping exactly 3 heads in 5 coin flips?

To start, we must understand that each coin flip is a Bernoulli trial, meaning it has two possible outcomes: heads or tails. The probability of getting a head in a single flip is 0.5, and similarly, the probability of getting a tail is also 0.5, since we're assuming a fair coin.

When we're asked about the probability of flipping exactly 3 heads out of 5 flips, we're delving into the realm of binomial probability. The binomial distribution is perfect for this scenario because it describes the number of successes in a fixed number of independent Bernoulli trials, given a constant success probability.

To calculate this probability, we use the binomial formula, which is:

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

Where: - ( P(X = k) ) is the probability of having exactly ( k ) successes (in our case, flipping heads) out of ( n ) trials (coin flips). - ( \binom{n}{k} ) is the number of ways to choose ( k ) successes out of ( n ) trials, also known as the combination formula (\frac{n!}{k!(n-k)!}). - ( p ) is the probability of success on an individual trial (flipping a head). - ( 1-p ) is the probability of failure (flipping a tail). - ( n ) is the total number of trials. - ( k ) is the number of successful trials we're interested in.

In our scenario, ( n = 5 ) (since we have 5 flips), ( k = 3 ) (since we want exactly 3 heads), and ( p = 0.5 ) (the probability of getting a head in a single flip). Plugging these values into our formula gives us:

[ P(X = 3) = \binom{5}{3} \times (0.5)^3 \times (0.5)^{5-3} ]

[ P(X = 3) = \frac{5!}{3!(5-3)!} \times (0.5)^3 \times (0.5)^2 ]

[ P(X = 3) = 10 \times (0.5)^3 \times (0.5)^2 ]

[ P(X = 3) = 10 \times (0.125) \times (0.25) ]

[ P(X = 3) = 10 \times 0.03125 ]

[ P(X = 3) = 0.3125 ]

Therefore, the probability of flipping exactly 3 heads in 5 coin flips is 0.3125, or 31.25%.

In my work as a Data Scientist, breaking down complex problems into solvable components and applying theoretical knowledge to real-world scenarios has been key. This approach not only provides clarity but also instills confidence in stakeholders when making data-driven decisions. By leveraging mathematical and statistical foundations, we can navigate through uncertainty with precision, bringing invaluable insights to the surface. This methodology, applied consistently across various projects, has empowered our teams to tackle challenges head-on, ensuring that our strategies are both robust and effective.