Conditional probability from tree

5 At a zoo, rides are offered on elephants, camels and jungle tractors. Ravi has money for only one ride. To decide which ride to choose, he tosses a fair coin twice. If he gets 2 heads he will go on the elephant ride, if he gets 2 tails he will go on the camel ride and if he gets 1 of each he will go on the jungle tractor ride.

1. Find the probabilities that he goes on each of the three rides. The probabilities that Ravi is frightened on each of the rides are as follows: $$\text { elephant ride } \frac { 6 } { 10 } , \quad \text { camel ride } \frac { 7 } { 10 } , \quad \text { jungle tractor ride } \frac { 8 } { 10 } .$$
2. Draw a fully labelled tree diagram showing the rides that Ravi could take and whether or not he is frightened. Ravi goes on a ride.
3. Find the probability that he is frightened.
4. Given that Ravi is not frightened, find the probability that he went on the camel ride.

3 It was found that \(68 \%\) of the passengers on a train used a cell phone during their train journey. Of those using a cell phone, \(70 \%\) were under 30 years old, \(25 \%\) were between 30 and 65 years old and the rest were over 65 years old. Of those not using a cell phone, \(26 \%\) were under 30 years old and \(64 \%\) were over 65 years old.

1. Draw a tree diagram to represent this information, giving all probabilities as decimals.
2. Given that one of the passengers is 45 years old, find the probability of this passenger using a cell phone during the journey.

7 Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel.
\includegraphics[max width=\textwidth, alt={}, center]{971a3594-3906-4868-b57c-e4667d42e5c8-3_712_1126_553_556} A day is selected at random. Find the probability that

1. the weather is wet and Andy travels by bus,
2. Andy walks or travels by bike,
3. the weather is dry given that Andy walks or travels by bike.

1 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, \(3 \%\) is classified as positive, \(6 \%\) as doubtful and the rest negative. In fact, of the people who test positive, only \(95 \%\) have the disease. Of the people who test doubtful, \(10 \%\) have the disease. Of the people who test negative, \(1 \%\) actually have the disease. People who do not have the disease are described as 'clear'.

1. Copy and complete the tree diagram to show this information.
   \includegraphics[max width=\textwidth, alt={}, center]{f3d936ba-8f60-4350-a5b3-92200996434c-1_833_1156_851_573}
2. Find the probability that a randomly selected person tests negative and is clear.
3. Find the probability that a randomly selected person has the disease.
4. Find the probability that a randomly selected person tests negative given that the person has the disease.
5. Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in \(98 \%\) of cases. If a person is clear, the examination will always correctly identify this.
6. A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.

7. Among the families with two children in a large city, the probability that the elder child is a boy is \(\frac { 5 } { 12 }\) and the probability that the younger child is a boy is \(\frac { 9 } { 16 }\). The probability that the younger child is a girl, given that the elder child is a girl, is \(\frac { 1 } { 4 }\).
One of the families is chosen at random. Using a tree diagram, or otherwise,

1. show that the probability that both children are boys is \(\frac { 1 } { 8 }\). Find the probability that
2. one child is a boy and the other is a girl,
3. one child is a boy given that the other is a girl. If three of the families are chosen at random,
4. find the probability that exactly two of the families have two boys.
5. State an assumption that you have made in answering part (d).

6. Serving against his regular opponent, a tennis player has a \(65 \%\) chance of getting his first serve in. If his first serve is in he then has a \(70 \%\) chance of winning the point but if his first serve is not in, he only has a \(45 \%\) chance of winning the point.

1. Represent this information on a tree diagram. For a point on which this player served to his regular opponent, find the probability that
2. he won the point,
3. his first serve went in given that he won the point,
4. his first serve didn't go in given that he lost the point.