Given that 65% of people who purchase sports cars are men. If 10 sports car owners are randomly selected, find the probability that exactly are men.
exactly what?
Exactly 6 are men.
To find the probability that exactly 6 out of 10 selected sports car owners are men, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting exactly k successes.
- n is the number of trials.
- p is the probability of success on a single trial.
- (n C k) is the combination formula.
In this case, n = 10 (10 sports car owners are randomly selected), p = 0.65 (probability that a randomly selected sports car owner is a man), and k = 6 (we want exactly 6 men out of the 10 selected).
The combination formula is given by:
(n C k) = n! / (k!(n-k)!)
Plugging in the values, we can calculate the probability:
P(X=6) = (10 C 6) * (0.65^6) * (1-0.65)^(10-6)
Let's calculate it step by step:
Step 1: Calculate n! (factorial of n):
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800
Step 2: Calculate k! (factorial of k):
6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
Step 3: Calculate (n-k)! (factorial of (n-k)):
(10-6)! = 4! = 4 * 3 * 2 * 1 = 24
Step 4: Calculate (n C k) using the combination formula:
(10 C 6) = 10! / (6! * (10-6)!)
= 3,628,800 / (720 * 24)
= 210
Step 5: Plug in the values into the binomial probability formula:
P(X=6) = (210) * (0.65^6) * (1-0.65)^(10-6)
Step 6: Calculate (0.65^6):
(0.65^6) = 0.090919
Step 7: Calculate (1-0.65)^(10-6):
(1-0.65)^(10-6) = 0.1296
Step 8: Calculate the final probability:
P(X=6) = (210) * 0.090919 * 0.1296
The final probability is the product of the values in Step 8:
P(X=6) ≈ 2.148
Therefore, the probability that exactly 6 out of 10 selected sports car owners are men is approximately 2.148.
To find the probability that exactly a certain number of sports car owners are men, we can use the binomial probability formula.
The binomial probability formula is:
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 or observations
k is the number of successes
p is the probability of success in a single trial
In this case, n = 10 (since we are selecting 10 sports car owners), k = ?, and p = 65% or 0.65.
Let's calculate the probability of exactly k men among the 10 selected sports car owners.
P(X = k) = (10Ck) * (0.65)^k * (1-0.65)^(10-k)
Since we want to find the probability that exactly k are men, we substitute k for the question mark.
P(X = k) = (10Ck) * (0.65)^k * (1-0.65)^(10-k)
Now, you can substitute the value of k (the number of men) into the formula to find the probability.