Exercise 8 Page 36

A student must choose one of the subjects, art, geology, or psychology, as an elective. She is equally likely to choose art or psychology and twice as likely to choose geology. What are the respective probabilities that she chooses art, geology, and psychology?

Solution

She is equally likely to choose one out of Art, Psychology or Geology, So the Probability = 1.

P(A) = Probability of choosing Art, P(P) = Probability of choosing Psychology, P(G) - Probability of choosing Geology

Thus, P(A) + P(P) + P(G) = 1

She is equally likely to choose art or psychology and twice as likely to choose geology. Since there is no overlap of probability

Let’s guess probability of Art is X, and Probability of Psychology is same as Art, and Probability of Geology is Twice than Art. \[ \begin{aligned} P(A) + P(P) + P(G) = 1 \\ X + X + 2X = 1 \\ 4X = 1 \\ X = \frac{1}{4} \end{aligned} \] \(P(A) = P(P) = X = \frac{1}{4}\) and, \(P(G)= 2X = 2\frac{1}{4} = \frac{1}{2}\)

Exercise 9 Page 36

A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability 5/8, French with probability 5/8, and art and French together with probability 1/4. What is the probability that he chooses mathematics? What is the probability that he chooses either art or French?

Solution

The information we have

He chooses art with probability 5/8 So,

\(P(A) = P(A \cup F ) \cap P(A \cup M ) =\frac {5}{8}\)

French with probability 5/8 so

\(P(F) = P(A \cup F )\cap P(F \cup M ) =\frac {5}{8}\)

Art and French together with probability 1/4.

\(P(A \cup F )=\frac {1}{4}\)

He must choose exactly two out of 3. So, Probability of

\[ P(A \cup F ) + P(A \cup M ) + P(F \cup M ) = 1 \\ \frac {1}{4} + P(A \cup M ) + P(F \cup M ) = 1 \\ P(A \cup M ) + P(F \cup M ) = 1-\frac {1}{4} \\ P(A \cup M ) + P(F \cup M ) = \frac {3}{4} \] So, Probability of Mathematics is \(\frac {3}{4}\)

  1. What is the probability that he chooses either art or French? Answer is Clear he always have to pick either Art or French beacause He must have to pick 2 out of 3.

\[ \begin{aligned} P(A \cup F) &= P(A) + P(F) - P(A\cap F)\\ &= \frac{5}{8} + \frac{5}{8} - \frac{1}{4} \\ &= \frac{8}{8} = 1 \end{aligned} \]