Please help me solve this problem step by step pleaseeee!
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 probability of exactly r successes in n attempts, where p is the probability of success on a single attempt, and q is the probability of failure on a single attempt, is given by the term containing pr in the binomial expansion of (p + q)n.
In this case, p = 0.8 (probability of hitting the target) and q = 0.2 (probability of missing the target).
The binomial expansion formula is given by:
P(r successes in n attempts) = nCr * pr * q(n-r)
Where nCr represents "n choose r," also known as the binomial coefficient, which can be calculated using the formula:
nCr = n! / (r!(n-r)!)
Let's calculate the probability:
Step 1: Calculate the binomial coefficient (nCr):
n = 6 (number of attempts)
r = 4 (number of successes)
nCr = 6! / (4!(6-4)!)
= 6! / (4! * 2!)
= (6 * 5 * 4!)/(4! * 2)
= (6 * 5) / 2
= 30 / 2
= 15
Step 2: Calculate the probability of hitting the target exactly four times (pr):
p^r = 0.8^4
= 0.4096
Step 3: Calculate the probability of missing the target two times (q(n-r)):
q^(n-r) = 0.2^(6-4)
= 0.2^2
= 0.04
Step 4: Multiply the results from steps 1, 2, and 3 together:
P(4 successes in 6 attempts) = nCr * pr * q(n-r)
= 15 * 0.4096 * 0.04
= 0.24576
Therefore, the probability that the archer hits the target exactly four times in six attempts is 0.24576 or approximately 0.246.
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%.
well, they tell you that
the probability that the archer hits the target exactly r times in n attempts is given by the term containing p^r in the binomial expansion of (p+q)^n
You have p=0.8 and q=0.2
You want the term containing p^4 in the expansion of (p+q)^6
That term will be C(6,2)p^4q^2
= 15(.8^4)(.2^2) = 0.25