Find the probability that among six randomly selected people, exactly four of them have brown eyes. Assume that 40% of the population has brown eyes
p = .4 so (1-p) = .6
binomial distribution
n = 6
k = 4
C(n,k) = n!/[ k! (n-k)! ]
= 6!/[4!(2!)] = 6*5/ 2 = 15
P(n,k) = 15 * p^k * (1-p)^(n-k)
=15 * .4^4 * .6*2
= .1152
To find the probability that exactly four out of six randomly selected people have brown eyes, we can use the binomial probability formula.
The binomial probability formula is:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials
x is the number of successful outcomes
p is the probability of a successful outcome
C(n, x) is the number of ways to choose x items from a set of n items, which can be calculated using the binomial coefficient formula: C(n, x) = n! / (x! * (n - x)!)
In this case, the probability of a person having brown eyes is p = 0.40, and we want exactly four people out of six to have brown eyes, so x = 4 and n = 6.
Using the above information, we can calculate the probability as follows:
P(4) = C(6, 4) * (0.40)^4 * (1 - 0.40)^(6 - 4)
Calculating C(6, 4):
C(6, 4) = 6! / (4! * (6 - 4)!)
= 6! / (4! * 2!)
= (6 * 5 * 4!) / (4! * 2)
= (6 * 5) / 2
= 15
Now, plugging the values into the binomial probability formula:
P(4) = 15 * (0.40)^4 * (1 - 0.40)^(6 - 4)
= 15 * 0.40^4 * 0.60^2
= 15 * 0.0256 * 0.36
= 0.13824
Therefore, the probability that exactly four out of six randomly selected people have brown eyes is approximately 0.13824, or 13.824%.
To find the probability that exactly four out of six randomly selected people have brown eyes, we can use the binomial probability formula.
The binomial probability formula is given by: P(x) = (n choose x) * p^x * (1 - p)^(n - x),
where:
- P(x) is the probability of having exactly x successes,
- n is the total number of trials or events,
- p is the probability of success in a single trial or event, and
- (n choose x) is the binomial coefficient, which represents the number of ways to choose x successes from n trials.
In this case, we need to find the probability of exactly four out of the six people having brown eyes. So, let's plug in the values into the formula:
P(4) = (6 choose 4) * (0.40)^4 * (1 - 0.40)^(6 - 4).
Calculating each component:
(6 choose 4) = 6! / (4!(6-4)!) = (6 * 5 * 4!) / (4! * 2!) = 15,
(0.40)^4 = 0.0256,
(1 - 0.40)^(6 - 4) = 0.36^2 = 0.1296.
Now, substituting these values back into the formula:
P(4) = 15 * 0.0256 * 0.1296
P(4) ≈ 0.0492.
Therefore, the probability that exactly four out of six randomly selected people have brown eyes is approximately 0.0492 or 4.92%.