Answer :
(a) The probability that exactly 4 out of 12 randomly sampled 18 to 20 year old consumes an alcoholic drink is 0.008127.
(b) The probability that exactly 8 out of 12, 18 to 20-year-olds have not consumed an alcoholic beverage is 0.231991.
(c) The probability that at most two out of four randomly sampled 18 to 20 year old have consumed alcoholic beverages is 0.35.
(d) The probability that at least one out of four randomly sample 18 to 20 years has consumed alcoholic beverages is 0.99168.
What is probability?
The possibility of an event happening is defined by probability. We may need to forecast an event's result in a variety of real-world circumstances. The outcomes of an event may be certain or uncertain to us.
Given that 69.8% of people of 18-20 yr old consume alcohol.
Then p=0.698
Now let X is the random variable denoting the number of people who drinks alcohol.
(a) Let the sample size, n = 12, p = 0.698
It can be seen that X follows a binomial distribution
Then,
P(X=4) = ¹²C₄p⁴(1-p)⁸
= 495 × (0.698)⁴(1-0.698)⁸
=495 × 0.2373 × 0.00006919
= 0.008127
(b) P(X=8) = ¹²C₈p⁸(1-p)⁴
= 12!/ (8!)(4!)(0.698)⁸(1-0.698)⁴
=495 × 0.05634344 × 0.0083181
= 0.231991
(c) Let sample size n=4
p= 0.698
Then,
P(X≤2) = P(X=0)+P(X=1)+P(X=2)
= ⁴C₀p⁰(1-p)⁴ + ⁴C₁p¹(1-p)³ + ⁴C₂p²(1-p)²
=1 × 1× 0.0083181 + 4× 0.698 × 0.0275 + 6× 0.4872 × 0.091204
=0.3516881
=0.35
(d)
For P(X≥1)= 1-P(X<1)
=1-P(X=0)
=1-⁴C₀ p⁰(1-p)⁴
=1-1 × 1 × 0.0083181
= 0.99168
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