An insurance company reported that 50% of all automobile damage claims were made bye people under the age of 25. If seven automobile damage claims were selected at random, determine the probability that exactly five of them were made by someone under the age of 25.
To determine the probability of exactly five out of seven automobile damage claims being made by someone under the age of 25, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes (in this case, claims made by people under the age of 25).
n is the total number of trials (in this case, the number of claims selected, which is 7).
k is the number of successful trials (in this case, 5).
p is the probability of success (in this case, the probability of a claim being made by someone under the age of 25, which is 0.5).
C(n, k) is the number of combinations of n items taken k at a time (in this case, 7 choose 5).
Plugging in the values into the formula:
P(X = 5) = C(7, 5) * 0.5^5 * (1-0.5)^(7-5)
Calculating the values:
C(7, 5) = 7! / (5! * (7-5)!) = 21
0.5^5 = 0.03125
(1-0.5)^(7-5) = 0.25
P(X = 5) = 21 * 0.03125 * 0.25 = 0.1640625
Therefore, the probability that exactly five out of seven automobile damage claims were made by someone under the age of 25 is 0.1640625, or approximately 16.41%.
To determine the probability of exactly five out of seven automobile damage claims being made by someone under the age of 25, 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 getting exactly k successes
- n is the total number of trials (in this case, seven automobile damage claims)
- k is the number of successes (in this case, five automobile damage claims made by someone under the age of 25)
- p is the probability of success on a single trial (in this case, 50% or 0.5)
- (nCk) is the binomial coefficient, also known as "n choose k", which represents the number of combinations of choosing k items from a set of n items.
Using the formula, we can calculate the probability as follows:
P(X = 5) = (7C5) * (0.5)^5 * (1-0.5)^(7-5)
First, let's calculate the binomial coefficient:
(7C5) = 7! / (5! * (7-5)!)
= (7 * 6 * 5!) / (5! * 2!)
= (7 * 6) / 2
= 21
Next, let's substitute the values into the formula and calculate the probability:
P(X = 5) = 21 * (0.5)^5 * (0.5)^(7-5)
= 21 * 0.5^5 * 0.5^2
= 21 * 0.03125 * 0.25
= 21 * 0.0078125
= 0.1640625
Therefore, the probability that exactly five out of seven automobile damage claims were made by someone under the age of 25 is approximately 0.164 or 16.4%.
prob(under 25) = .5
prob(not under 25) = .5
prob(exactly 5 of 7 are under 25) = C(7,5)(.5)^5 (.5)^2
= 21(.5)^7 = .....