Suppose that
30%
of all college students smoke cigarettes. A sample of
18
is selected randomly. What is the probability that exactly
10
students smoke? Round your answer to four decimal places.
http://www.statisticshowto.com/binomial-distribution-formula/
To calculate the probability that exactly 10 students out of 18 smoke cigarettes, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of exactly k successes,
n is the total number of trials (sample size),
k is the number of successful trials (students who smoke),
p is the probability of success (percentage of college students who smoke), and
nCk is the combination formula.
In this case, n = 18, k = 10, and p = 0.3 (30%).
Let's calculate the probability:
P(X = 10) = (18C10) * 0.3^10 * (1-0.3)^(18-10)
Using the combination formula nCk, which is calculated as:
nCk = (n!)/((n-k)! * k!)
P(X = 10) = (18!)/((18-10)! * 10!) * 0.3^10 * (1-0.3)^(18-10)
Calculating the combination:
18C10 = (18!)/((18-10)! * 10!)
Now let's calculate this:
18C10 = (18!)/((18-10)! * 10!)
= (18!)/(8! * 10!)
= 43758
Substituting back into the probability formula:
P(X = 10) = 43758 * 0.3^10 * 0.7^8
Now let's calculate this probability:
P(X = 10) = 43758 * 0.00282475249 * 0.05764801
= 0.7094 (rounded to four decimal places)
Therefore, the probability that exactly 10 students out of 18 smoke cigarettes is approximately 0.7094.
To find the probability that exactly 10 students smoke cigarettes out of a sample of 18, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes,
n is the number of trials or observations,
x is the number of desired successes,
p is the probability of success in a single trial,
(1-p) is the probability of failure in a single trial,
and (nCx) is the number of combinations of n items taken x at a time.
In this case:
n = 18 (sample size)
x = 10 (number of students smoking)
p = 0.30 (probability of a college student smoking)
Now, let's calculate the probability using the formula:
P(10) = (18C10) * (0.30)^10 * (1-0.30)^(18-10)
To calculate (18C10), we use the combination formula:
(18C10) = 18! / (10! * (18-10)!)
Note: The symbol "!" represents the factorial of a number, which means multiplying all positive integers from 1 to that number.
Let's calculate (18C10) first:
(18C10) = 18! / (10! * 8!)
Using factorials, we simplify calculations:
(18C10) = (18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10!) / (10! * 8!)
The 10! cancels out in numerator and denominator:
(18C10) = (18 * 17 * 16 * 15 * 14 * 13 * 12 * 11) / 8!
Calculating the numerator:
(18C10) = 243,756
Now, we can substitute the values in the original formula:
P(10) = (18C10) * (0.30)^10 * (1-0.30)^(18-10)
P(10) = 243,756 * (0.30)^10 * (0.70)^8
Calculating (0.30)^10:
(0.30)^10 ≈ 0.0000059049
Calculating (0.70)^8:
(0.70)^8 ≈ 0.057648
Now, we can substitute these values back into the formula:
P(10) = 243,756 * 0.0000059049 * 0.057648
Calculating this expression:
P(10) ≈ 0.0017336
Therefore, the probability that exactly 10 students smoke out of a sample of 18 is approximately 0.0017 when rounded to four decimal places.
http://onlinestatbook.com/2/probability/binomial.html
C(18,10) = 18!/[10!(18-10)]
P(18) = C(18,10) * .3^10 * .7*8