If 95% of all people between 21 and 50 have had at least one adult beverage in their life, find these probabilities for a sample of 10 people in that age group:
a) Exactly 10 have had an adult beverage
b) At least 6 have had an adult beverage
c) At most 8 have had an adult beverage
d) At most 1 has had an adult beverage.
Here are some properties of the experiment:
1. Each trial is Bernoulli (i.e. yes/no)
2. Number of trials is fixed.
3. probability of success does not vary from trial to trial.
4. each trial is independent of the others (assumed random sample).
All these point to the binomial distribution with N=10, p=0.95, q=(1-0.95)=0.05, and success="have had at least one adult beverage".
P(X=i)=(N,i)p^i q^(N-i)
and (N,i) = number of ways to choose i objects from N distinct objects.
(N,i)=N!/(i!(N-i)!)
a). N=10, i=10, p=0.95, q=0.05
P(X=10)=(10,10)0.95^10 0.05^0
=1*0.95^10*1
b) At least 6 => X=6,7,8,9,10
So
P(X≥6)=∑P(X=i), i=6 to 10
c) At most 8 => X=1,2,3...8
=> X≠9 and X≠10
We can use the identity
P(X≤8)=∑P(X=i), i=1 to 8
=1-P(X≥9)
=1- [∑P(X=i), i=9 to 10]
which simplifies the calculations.
d) at most 1 => X=0 or X=1.
To solve these probabilities, we will be using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- P(X=k) is the probability of exactly k successes,
- n is the total number of trials,
- k is the number of successful outcomes,
- p is the probability of success on any given trial.
In this case, n = 10 (sample size) and p = 0.95 (probability of having had an adult beverage).
Let's calculate the probabilities for each scenario:
a) Exactly 10 have had an adult beverage:
P(X=10) = (10 choose 10) * 0.95^10 * (1-0.95)^(10-10)
= 1 * 0.95^10 * 0.05^0
= 0.95^10
≈ 0.5987
b) At least 6 have had an adult beverage:
P(X>=6) = P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)
To calculate each individual probability, use the same binomial formula.
c) At most 8 have had an adult beverage:
P(X<=8) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8)
To calculate each individual probability, use the same binomial formula.
d) At most 1 has had an adult beverage:
P(X<=1) = P(X=0) + P(X=1)
To calculate each individual probability, use the same binomial formula.
By plugging in the values into the binomial probability formula, you can calculate the probabilities for each scenario. Remember to use the choose function (combination) to calculate the binomial coefficient (n choose k).