Respuesta :
Answer:
The probability that the senator was in the Democratic party, given that the senator was returning to office is 0.4715.
Step-by-step explanation:
The complete question is:
Sophia made the following two-way table categorizing the US senators in 2015 by their political party and whether or not it was their first term in the senate.
Democratic Republican Independent Total
First Term 11 28 11 50
Returning 33 26 11 70
Total 44 54 22 120
Find the probability that the senator was in the Democratic party, given that the senator was returning to office.
Solution:
The conditional probability of an event A given that another event X has already occurred is given by:
[tex]P(A|X)=\frac{P(A\cap X)}{P(X)}[/tex]
The probability of an event E is given by the ratio of the number of favorable outcomes to the total number of outcomes.
[tex]P(E)=\frac{n(E)}{N}[/tex]
Compute the probability of selecting an US senator who is a Democratic and was returning to office as follows:
[tex]P(D\cap R)=\frac{33}{120}=0.275[/tex]
Compute the probability of selecting an US senator who was returning to office as follows:
[tex]P(R)=\frac{70}{120}=0.5833[/tex]
Compute the conditional probability, P (D | R) as follows:
[tex]P(D|R)=\frac{P(D\cap R)}{P(R)}[/tex]
[tex]=\frac{0.275}{0.5833}\\\\=0.4714555\\\\\approx 0.4715[/tex]
Thus, the probability that the senator was in the Democratic party, given that the senator was returning to office is 0.4715.
