Question

Birth certificates show that approximately 9% of all births in the United States are to teen mothers (ages 15 to 19), 24% to young‑adult mothers (age 20 to 24), and the remaining 67% to adult mothers (age 25 to 44). An extensive survey of live births examined pregnancy type, defining an unintended pregnancy as one that was unwanted or mistimed by at least two years. The survey found that "only 23% of births to teen mothers are intended, and 77% are unintended. Among births to young‑adult women age 20 – 24, 50% are intended, and at ages 25 – 44, 75% are intended."Use your tree diagram and Bayes’s theorem to find the probability that a birth was to a teen mother if we know that the pregnancy was unintended.

Answer 1

Answer:

The probability that a birth was to a teen mother if we know that the pregnancy was unintended is 0.1942.

Step-by-step explanation:

Denote the events as follows:

T = birth was to a teen mother

Y = birth was to a young-adult mother

A = birth was to an adult mother

U = a birth was unintended

I = a birth was intended.

Given:

P (T) = 0.09

P (Y) = 0.24

P (A) = 0.67

P (I | T) = 0.23

P (U | T) = 0.77

P (I | Y) = 0.50

P (U | Y) = 1 - P (I | Y) = 0.50

P (I | A) = 0.75

P (U | A) = 1 - P (I | A) = 0.75 = 0.25

Consider the tree diagram below.

According to the Bayes' theorem,

$P(B|A)=\frac{P(A|B)P(B)}{P(A|B)P(B)+P(A|B^{c})P(B^{c})}$

Using the Bayes' theorem compute the probability that a birth was to a teen mother if we know that the pregnancy was unintended as follows:

$P(T|U)=\frac{P(U|T)P(T)}{P(U|T)P(T)+P(U|Y)P(Y)+P(U|A)P(A)} \\=\frac{(0.77\times0.09)}{(0.77\times0.09)+(0.50\times0.24)+(0.25\times0.67)} \\=0.19423\\\approx0.1942$

The probability that a birth was to a teen mother if we know that the pregnancy was unintended is 0.1942.