# At a High school, 70% of the seniors have taken an advance calculus course. Of those who have taken Advanced Calculus, 60% will apply for a Pre-Health

At a High school, 70% of the seniors have taken an advance calculus course. Of those who have taken Advanced Calculus, 60% will apply for a Pre-Health science major when they apply for college admission. Of those who have not taken advanced calculus, 40% will apply for a Pre-Health science major when they apply for college admission. A senior is selected at random. What is the probability that the senior have taken advanced calculus, given that the senior will apply for a pre-health science major?
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Step 1
Given data:
The given percentage of the students that are taken an advanced calculus course is P(A)=70%=0.70.
The given percentage of the students who have taken Advanced calculus and applied for pre-health science is $P\left(A\cap H\right)=60\mathrm{%}P\left(A\right)$.
The given percentage of the students who have not taken Advanced calculus and applied for pre-health science is P(B)=40%(100%-P(A)).
The given percentage of the students who have taken Advanced calculus and applied for pre-health science is,
$P\left(A\cap H\right)=60\mathrm{%}\left(0.70\right)=0.42$
The given percentage of the students who have not taken Advanced calculus and applied for pre-health science is,
P(B)=40%(1-0.7)=0.12
Step 2
The expression for the probability for the students who are taken pre-health science is, $P\left(H\right)=P\left(A\cap H\right)+P\left(B\right)$
Substitute the given values in the above expression.
P(H)=0.42+0.12=0.54
The expression for the probability that the senior has taken advanced-calculus given that senior applied for pre-health science is,
$P\left(\frac{A}{H}\right)=\frac{P\left(A\cap H\right)}{P\left(H\right)}$
Substitute the above-calculated values in the expression.
$P\left(\frac{A}{H}\right)=\frac{0.42}{0.54}=0.777$
Thus, the probability that the senior has taken advanced-calculus given that the senior applied for pre-health science is 0.777.