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

From the Tree Diagram:

Probability the randomly selected mother is teen mother : P(T) = 0.09

Probability the randomly selected mother is young adult mother : P(Y) = 0.24

Probability the randomly selected mother is an adult mother : P(A) = 0.63

U : Event of pregnancy was unintended

Probability that the pregnancy was unintended given that  birth was to a teen mother = P(U|T) = 0.77

Probability that the pregnancy was unintended given that  birth was to a young adult mother = P(U|Y) = 0.50

Probability that the pregnancy was unintended given that  birth was to an adult mother = P(U|A) = 0.25

Probability that a birth was to a teen mother if we know that the pregnancy was unintended = P(T|U)

By Bayes theorem,

P(T)P(U|T) = 0.09 x 0.77 = 0.0693

P(Y)P(U|Y) = 0.24 x 0.50 = 0.12

P(A)P(U|A) = 0.67 x 0.25 = 0.1675

P(T)P(U|T)+P(Y)P(U|Y)+P(A)P(A|T) = 0.0693+0.12+0.1675=0.3568

Probability that a birth was to a teen mother if we know that the pregnancy was unintended = 0.194226457