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 the probability of getting r successes in n attempts, when the probability of success is p, and the probability of failure is q, is given by the term containing pr in the binomial expansion of (p + q)n.
In this case, the probability of hitting the target (success) is p = 0.8, and the probability of missing the target (failure) is q = 0.2.
Using the binomial expansion formula, the probability of hitting the target exactly four times in six attempts is given by the term containing p^4 in the expansion of (p + q)^6.
Here's how we can calculate it step by step:
Step 1: Calculate the probability of hitting the target exactly four times, which is p^4. In this case, it is (0.8)^4.
Step 2: Calculate the probability of missing the target exactly two times, which is q^2. In this case, it is (0.2)^2.
Step 3: Calculate the number of ways we can arrange four hits and two misses among the six attempts. This is given by the binomial coefficient (6 choose 4) or C(6, 4), which represents the number of combinations. C(6, 4) is calculated as 6! / (4! * (6-4)!), where ! denotes factorial.
Step 4: Finally, multiply the probabilities calculated in steps 1, 2, and 3 to find the probability of hitting the target exactly four times in six attempts.
Probability = (p^4) * (q^2) * C(6, 4)
Now, let's plug in the values and calculate the probability:
Probability = (0.8)^4 * (0.2)^2 * (6! / (4! * 2!))
Probability = (0.4096) * (0.04) * (6! / (4! * 2!))
Probability = 0.0656 * 0.04 * (720 / (24 * 2))
Probability = 0.0656 * 0.04 * 30
Probability = 0.07872
So, the probability that the archer hits the target exactly four times in six attempts is approximately 0.07872 or about 7.872%.