Final answer:
The probability of getting exactly 5 heads when a coin is flipped 8 times, using the binomial probability formula, is approximately 0.219 (rounded to three decimal places), which corresponds to option (c).
Step-by-step explanation:
To calculate the probability of getting exactly 5 heads when a coin is flipped 8 times, we can use the binomial probability formula:
The binomial probability formula is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting k successes in n trials.
- C(n, k) is the number of combinations of n things 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, the success is flipping heads, which has a probability p = 0.5, since a fair coin has two equally likely outcomes. The number of trials n = 8, and we're looking for k = 5 heads. Using the formula, we get:
P(X = 5) = C(8, 5) * (0.5)^5 * (0.5)^(8-5)
Calculating the combination C(8, 5), we get 56. Thus:
P(X = 5) = 56 * (0.5)^5 * (0.5)^3 = 56 * (0.5)^8
P(X = 5) = 56 * 0.00390625 = 0.21875
Therefore, the probability of the event where exactly 5 heads appear when flipping a coin 8 times is 0.21875, which can be approximated as 0.219, the option (c).