Can anybody help me on this problem for my Algebra 2 class, I am having a hard time tyring to solve it step by step.
The probability that an archer hits the target is
p = 0.8, so the probability that he misses the target is q = 0.2. It is known that in this situation the probability that the archer hits the target exactly r times in n attempts is given by the term containing pr in the binomial expansion of (p + q)^n.
Find the probability that the archer hits the target exactly four times in six attempts
To find the probability that the archer hits the target exactly four times in six attempts, we can use the binomial expansion formula.
The formula for the binomial expansion states that the probability of getting r successes in n attempts is given by:
P(r) = (nCr) * p^r * q^(n-r)
Where:
- P(r) is the probability of getting r successes
- nCr represents "n choose r," which is the number of ways to choose r objects from a set of n distinct objects. It is calculated as nCr = n! / (r! * (n-r)!), where "!" denotes the factorial of a number.
- p is the probability of success in a single attempt
- q is the probability of failure (or missing the target) in a single attempt
In this problem, we want to find the probability of hitting the target exactly four times in six attempts. So we have:
r = 4 (number of successes)
n = 6 (number of total attempts)
p = 0.8 (probability of success)
q = 0.2 (probability of failure)
Now, let's plug these values into the formula:
P(4) = (6C4) * 0.8^4 * 0.2^(6-4)
To calculate (6C4), we need to calculate the factorials of 6, 4, and (6-4):
6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
4! = 4 * 3 * 2 * 1 = 24
(6-4)! = 2!
Now, substitute these values back into the formula:
P(4) = (720) / (24 * 2!) * 0.8^4 * 0.2^2
Simplifying further:
P(4) = (720) / (24 * 2) * 0.8^4 * 0.2^2
P(4) = 15 * 0.8^4 * 0.2^2
Calculate the exponential powers:
P(4) = 15 * (0.4096) * (0.04)
P(4) = 15 * 0.016384
P(4) = 0.24576
Therefore, the probability that the archer hits the target exactly four times in six attempts is approximately 0.24576 or 24.576%.