Answer:
A) 10.678%
B) 37.336%
Step-by-step explanation:
To find the probability that a certain number of the 50 students had type A blood, given that 31% of the US population has type A blood, we can model the given scenario as a binomial distribution.
Binomial distribution
[tex]\large\boxed{X\sim \text{B}(n,p)}[/tex]
where:
Since we want to find the probability of students having type A blood, we consider "type A" as the success and "type AB" as the failure, so we have:
Therefore:
[tex]\large\boxed{X\sim \text{B}(50,0.31)}[/tex]
where the random variable X represents the number of students with type A blood.
To find the probability that exactly 17 of the students had type A blood, we need to find P(X = 17).
To do this, we can either use the binomial probability formula, or enter the values into a statistical calculator.
Using the binomial probability formula, we get:
[tex]\displaystyle P(X = 17) = \binom{50}{17} \times 0.31^{17} \times (1 - 0.31)^{50 - 17}[/tex]
[tex]P(X = 17) = 0.106779887...[/tex]
[tex]P(X = 17) = 10.678\%\;\sf(3\;d.p.)[/tex]
Therefore, the probability that exactly 17 of the students had type A blood is 10.678% (rounded to three decimal places).
To find the probability that at least 17 of the students had type A blood, we need to find P(X ≥ 17).
To do this, we can use the binomial cumulative distribution function on a calculator. As the calculator function gives us P(X ≤ x) for X ~ B(n, p), we will need to calculate P(X ≤ 16) and then subtract it from 1:
[tex]P(X \geq 17) = 1 - P(X \leq 16)[/tex]
[tex]P(X \geq 17) = 1 - 0.626641773...[/tex]
[tex]P(X \geq 17) = 0.373358226...[/tex]
[tex]P(X \geq 17) = 37.336\%\;\sf(3\;d.p.)[/tex]
Therefore, the probability that at least 17 of the students had type A blood is 37.336% (rounded to three decimal places).
