EXAMPLE 4.47

Online chat rooms. Online chat rooms are dominated by the young. Teens are the biggest users. If we look only at adult Internet users (aged 18 and over), 47% of the 18 to 29 age group chat, as do 21% of the 30 to 49 age group and just 7% of those 50 and over. To learn what percent of all Internet users participate in chat, we also need the age breakdown of users. Here it is: 29% of adult Internet users are 18 to 29 years old (event A1), another 47% are 30 to 49 (event A2), and the remaining 24% are 50 and over (event A3).

What is the probability that a randomly chosen adult user of the Internet participates in chat rooms (event C)? To find out, use the tree diagramtree diagram in Figure 4.19 to organize your thinking. Each segment in the tree is one stage of the problem. Each complete branch shows a path through the two stages. The probability written on each segment is the conditional probability of an Internet user following that segment, given that he or she has reached the node from which it branches.

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Figure 4.19 Tree diagram, Example 4.47. The probability P(C) is the sum of the probabilities of the three branches marked with asterisks (*).

Starting at the left, an Internet user falls into one of the three age groups. The probabilities of these groups

P(A1) = 0.29  P(A2) = 0.47  P(A3) = 0.24

mark the leftmost branches in the tree. Conditional on being 18 to 29 years old, the probability of participating in chat is P(C | A1) = 0.47. So the conditional probability of not participating is

P(Cc | A1) = 1 − 0.47 = 0.53

These conditional probabilities mark the paths branching out from the A1 node in Figure 4.19. The other two age group nodes similarly lead to two branches marked with the conditional probabilities of chatting or not. The probabilities on the branches from any node add to 1 because they cover all possibilities, given that this node was reached.

There are three disjoint paths to C, one for each age group. By the addition rule, P(C) is the sum of their probabilities. The probability of reaching C through the 18 to 29 age group is

P(C and A1) = P(A1)P(C | A1)

= 0.29 × 0.47 = 0.1363

Follow the paths to C through the other two age groups. The probabilities of these paths are

P(C and A2) = P(A2)P(C | A2) = (0.47)(0.21) = 0.0987

P(C and A3) = P(A3)P(C | A3) = (0.24)(0.07) = 0.0168

The final result is

P(C) = 0.1363 + 0.0987 + 0.0168 = 0.2518

About 25% of all adult Internet users take part in chat rooms.