Tree diagram with two-stage events

1 Juan goes to college each day by any one of car or bus or walking. The probability that he goes by car is 0.2 , the probability that he goes by bus is 0.45 and the probability that he walks is 0.35 . When Juan goes by car, the probability that he arrives early is 0.6 . When he goes by bus, the probability that he arrives early is 0.1 . When he walks he always arrives early.

1. Draw a fully labelled tree diagram to represent this information.
2. Find the probability that Juan goes to college by car given that he arrives early.

1 Fabio drinks coffee each morning. He chooses Americano, Cappucino or Latte with probabilities 0.5, 0.3 and 0.2 respectively. If he chooses Americano he either drinks it immediately with probability 0.8 , or leaves it to drink later. If he chooses Cappucino he either drinks it immediately with probability 0.6 , or leaves it to drink later. If he chooses Latte he either drinks it immediately with probability 0.1 , or leaves it to drink later.

1. Find the probability that Fabio chooses Americano and leaves it to drink later.
2. Fabio drinks his coffee immediately. Find the probability that he chose Latte.

2 Xavier, Yuri and Zara attend a sports centre for their judo club's practice sessions. The probabilities of them arriving late are, independently, \(0.3,0.4\) and 0.2 respectively.

1. Calculate the probability that for a particular practice session:
   1. all three arrive late;
   2. none of the three arrives late;
   3. only Zara arrives late.
2. Zara's friend, Wei, also attends the club's practice sessions. The probability that Wei arrives late is 0.9 when Zara arrives late, and is 0.25 when Zara does not arrive late. Calculate the probability that for a particular practice session:
   1. both Zara and Wei arrive late;
   2. either Zara or Wei, but not both, arrives late.

5 Hugh owns a small farm.

1. He has two sows, Josie and Rosie, which he feeds at a trough in their field at 8.00 am each day. The probability that Josie is waiting at the trough at 8.00 am on any given day is 0.90 . The probability that Rosie is waiting at the trough at 8.00 am on any given day is 0.70 when Josie is waiting at the trough, but is only 0.20 when Josie is not waiting at the trough. Calculate the probability that, at 8.00 am on a given day:
   1. both sows are waiting at the trough;
   2. neither sow is waiting at the trough;
   3. at least one sow is waiting at the trough.
2. Hugh also has two cows, Daisy and Maisy. Each day at 4.00 pm , he collects them from the gate to their field and takes them to be milked. The probability, \(\mathrm { P } ( D )\), that Daisy is waiting at the gate at 4.00 pm on any given day is 0.75 .
   The probability, \(\mathrm { P } ( M )\), that Maisy is waiting at the gate at 4.00 pm on any given day is 0.60 .
   The probability that both Daisy and Maisy are waiting at the gate at 4.00 pm on any given day is 0.40 .
   1. In the table of probabilities, \(D ^ { \prime }\) and \(M ^ { \prime }\) denote the events 'not \(D\) ' and 'not \(M\) ' respectively.

5 Alison is a member of a tenpin bowling club which meets at a bowling alley on Wednesday and Thursday evenings. The probability that she bowls on a Wednesday evening is 0.90 . Independently, the probability that she bowls on a Thursday evening is 0.95 .

1. Calculate the probability that, during a particular week, Alison bowls on:
   1. two evenings;
   2. exactly one evening.
2. David, a friend of Alison, is a member of the same club. The probability that he bowls on a Wednesday evening, given that Alison bowls on that evening, is 0.80 . The probability that he bowls on a Wednesday evening, given that Alison does not bowl on that evening, is 0.15 . The probability that he bowls on a Thursday evening, given that Alison bowls on that evening, is 1 . The probability that he bowls on a Thursday evening, given that Alison does not bowl on that evening, is 0 . Calculate the probability that, during a particular week:
   1. Alison and David bowl on a Wednesday evening;
   2. Alison and David bowl on both evenings;
   3. Alison, but not David, bowls on a Thursday evening;
   4. neither bowls on either evening.

6. The values of the two variables \(A\) and \(B\) given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled \(A\) and \(B\). One card is selected from Pack A. Let \(A _ { i }\) represent the event that the first digit on this card is \(i\).

1. Write down the value of \(\mathrm { P } \left( A _ { 2 } \right)\). The card taken from Pack \(A\) is now transferred into Pack \(B\), and one card is picked at random from Pack \(B\). Let \(B _ { i }\) represent the event that the first digit on this card is \(i\).
2. Show that \(\mathrm { P } \left( A _ { 1 } \cap B _ { 1 } \right) = \frac { 1 } { 24 }\).
3. Show that \(\mathrm { P } \left( A _ { 6 } \mid B _ { 5 } \right) = \frac { 4 } { 41 }\).
4. Find the value of \(\mathrm { P } \left( A _ { 1 } \cup B _ { 3 } \right)\).

1. The following incomplete tree diagram shows the relationships between the event \(A\) and the event \(B\).
   \includegraphics[max width=\textwidth, alt={}, center]{77ae01cd-2b58-48ab-889f-272e27ecf99d-14_799_839_351_548}

Given that \(\mathrm { P } ( B ) = \frac { 9 } { 20 }\)

1. find \(\mathrm { P } ( A )\) and complete the tree diagram,
2. find \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\).