To solve the problem of finding the probability that more than 4 of the flips turn up tails when a fair coin is flipped 13 times, we need to use the binomial distribution. The binomial distribution formula is:

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

Where:

• $n = 13$ is the number of trials (flips),
• $k$ is the number of tails,
• $p = \frac{1}{2} = 0.5$ is the probability of getting tails on a single flip.

We are asked to find the probability that the number of tails $X$ is more than 4, i.e., $P(X > 4)$.

The probability can be written as:

$P(X > 4) = 1 - P(X \leq 4)$

Now, we calculate the cumulative probability $P(X \leq 4)$, which is the sum of the probabilities for 0, 1, 2, 3, and 4 tails.

Let me calculate it for you.The probability that more than 4 of the flips turn up tails is approximately 0.8666, which matches the provided option 0.866577.

Would you like more details on the binomial distribution or on how this result was calculated?

Here are 5 related questions to expand your understanding:

1. How is the binomial distribution used in probability theory?
2. How do you calculate the cumulative distribution for a binomial random variable?
3. What is the difference between the binomial distribution and the normal distribution?
4. How would the probability change if the coin was biased?
5. Can you use a normal approximation to estimate the binomial probability?

Tip: When dealing with binomial probabilities, cumulative probabilities are often easier to compute by subtracting unwanted outcomes from 1.