To find the probability that the archer hits the target exactly four times in six attempts, we can use the binomial expansion formula.
The binomial expansion formula states that for any integers n and k, the term containing pr in the binomial expansion of (p + q)^n is given by the formula:
C(n, r) * p^r * q^(n-r),
where C(n, r) represents the number of combinations of choosing r items from a set of n items, and can be calculated using the formula:
C(n, r) = n! / (r! * (n-r)!),
where ! denotes the factorial of a number.
In this question, we have n = 6 (6 attempts) and r = 4 (exactly four hits).
Let's calculate the probability step by step:
1. Calculate C(6, 4):
C(6, 4) = 6! / (4! * (6-4)!)
= 6! / (4! * 2!)
= (6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (2 * 1))
= (6 * 5) / (2 * 1)
= 15.
2. Calculate p^r:
p^r = 0.8^4 = 0.8 * 0.8 * 0.8 * 0.8 = 0.4096.
3. Calculate q^(n-r):
q^(n-r) = 0.2^(6-4) = 0.2^2 = 0.04.
4. Multiply all the calculated values:
C(6, 4) * p^r * q^(n-r) = 15 * 0.4096 * 0.04
= 15 * 0.016384
= 0.24576.
Therefore, the probability that the archer hits the target exactly four times in six attempts is approximately 0.24576, or 24.576%.