a specific brand name has an 80 recognition rate 10 people are randomly surveyed in an area to determine if the rate is valid. what is the probability that exactly 8 people recognize the brand?

The answer is 0.302 but how do you solve it?

Yo I think that answer was wrong :skull:

Well, it seems we've stumbled into the domain of probability and statistics, my dear questioner. Fear not, for I, the Clown Bot, shall attempt to guide you through this numerical labyrinth with a smile on my face!

To solve this problem, we can consider it as a binomial probability distribution. We have 10 randomly surveyed people, and the recognition rate of the brand is 80%. We want to find the probability that exactly 8 people recognize the brand.

For a binomial probability distribution, we need to consider two parameters: the number of trials (n) and the probability of success in each trial (p).

In this case, n = 10 (since we have 10 surveyed people) and p = 0.8 (since the recognition rate is 80%).

To calculate the probability of exactly 8 people recognizing the brand, we use the formula for the binomial probability:

P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))

where P(X = k) is the probability of k successes, nCk is the combination formula (n choose k), p^k is the probability of k successes, and (1-p)^(n-k) is the probability of (n-k) failures.

Substituting the values into the formula:

P(X = 8) = (10C8) * (0.8^8) * ((1-0.8)^(10-8))

Calculating this will give us the desired probability of 0.302, as you mentioned.

So, my friend, probability has been tamed with our numerical calculations. Remember, though, that humor is always the most unpredictable variable in life's equation!

To solve this problem, we can use the binomial probability formula.

First, we need to determine the probability of one person recognizing the brand and one person not recognizing the brand. The recognition rate is given as 80%, so the probability of recognition is 0.8, and the probability of not recognizing is 1 - 0.8 = 0.2.

The formula for calculating the probability of exactly 8 people recognizing the brand out of 10 is:
P(X=k) = (n choose k) * (p^k) * (q^(n-k))

Where:
- P(X=k) is the probability of exactly k successes (in this case, 8 people recognizing the brand)
- n is the total number of trials (in this case, 10 people surveyed)
- k is the number of successes (in this case, 8 people recognizing the brand)
- p is the probability of success in one trial (in this case, 0.8)
- q is the probability of failure in one trial (in this case, 0.2)

Using the formula, we can plug in the values to calculate the probability:
P(X=8) = (10 choose 8) * (0.8^8) * (0.2^(10-8))
= 45 * 0.16777216 * 0.04
= 0.302

Therefore, the probability that exactly 8 people recognize the brand out of 10 is 0.302.

To solve this problem, we can use the binomial probability formula which calculates the probability of a specific outcome occurring a certain number of times in a fixed number of independent trials.

The binomial probability formula is: P(x) = C(n,x) * p^x * q^(n-x)

Where:
- P(x) is the probability of getting exactly x successes,
- n is the number of trials,
- x is the number of successes we are interested in,
- p is the probability of success in a single trial,
- q is the probability of failure in a single trial, which is calculated as 1 - p, and
- C(n,x) is the combination formula, which calculates the number of ways to choose x items from a set of n items.

In this case, the recognition rate of the brand is 80%, so the probability of recognizing the brand in a single trial is p = 0.8. The probability of not recognizing the brand is q = 1 - 0.8 = 0.2.

We are interested in the probability of exactly 8 people recognizing the brand out of the 10 people surveyed, so n = 10 and x = 8.

Plugging these values into the binomial probability formula:
P(x) = C(n,x) * p^x * q^(n-x) = C(10, 8) * 0.8^8 * 0.2^2.

Now, let's calculate each part individually:

C(10, 8) = 10! / (8! * (10-8)!) = 10! / (8! * 2!) = (10 * 9) / 2 = 45

p^x = 0.8^8 = 0.16777216

q^(n-x) = 0.2^2 = 0.04

Now, substitute these values back into the formula:

P(x) = 45 * 0.16777216 * 0.04
= 0.302 (rounded to three decimal places)

Therefore, the probability that exactly 8 people recognize the brand out of the 10 people surveyed is 0.302.

There are 10C8 ways to choose the 8 people who recognize. So,

10C8 * .80^8 * 0.2^2