Question Type 3: Given a worded problem, converting to a tree diagram to calculate probability of an outcome Exercises

In a two-stage process, a person chooses Yes or No at the first stage with $P(\text{Yes}) = 0.5$ and $P(\text{No}) = 0.5$. If they choose Yes, then at the second stage $P(\text{Yes} \mid \text{Yes}) = 0.75$ and $P(\text{No} \mid \text{Yes}) = 0.25$. If they choose No at the first stage, then at the second stage $P(\text{Yes} \mid \text{No}) = 0.25$ and $P(\text{No} \mid \text{No}) = 0.75$. Calculate the probability of the outcome Yes then No.

A three-stage conditional decision process is defined as follows:

• The probability of 'Yes' at the first stage is $0.5$.
• Given 'Yes' at the first stage, the probability of 'Yes' at the second stage is $0.75$.
• Given 'Yes' at the first two stages, the probability of 'Yes' at the third stage is $0.8$.

Calculate the probability that there is at least one 'No' among the three decisions.

The process is extended to a third stage. After the second decision:

• If the second outcome is 'Yes', $P(\text{Yes} \mid \text{Yes}_2) = 0.8$ and $P(\text{No} \mid \text{Yes}_2) = 0.2$.
• If the second outcome is 'No', $P(\text{Yes} \mid \text{No}_2) = 0.3$ and $P(\text{No} \mid \text{No}_2) = 0.7$.

Find the probability of the sequence: Yes at stage 1, No at stage 2, Yes at stage 3.

Using the three-stage process described, calculate the probability that exactly two of the three decisions are No.

Calculate $P(\text{Yes}, \text{Yes}, \text{Yes})$.

Calculate the probability of the outcome 'Yes' then 'Yes'.

A student answers two True/False questions. The probability of answering the first correctly is $0.5$. If they answer the first correctly, the probability of answering the second correctly is $0.8$; if they answer the first incorrectly, the probability of answering the second correctly is $0.3$. Draw a tree diagram for this process and calculate the probability that the student answers exactly one question correctly.

A bag contains red and blue marbles. A marble is drawn at random: $P(\text{Red})=0.4$, $P(\text{Blue})=0.6$.

If a red marble is drawn first, the probabilities for the second draw are $P(\text{Red})=0.7$, $P(\text{Blue})=0.3$.

If a blue marble is drawn first, the probabilities for the second draw are $P(\text{Red})=0.2$, $P(\text{Blue})=0.8$.

Calculate the probability of drawing Red then Blue.

Consider a two-stage process where the probability of the outcome 'No' at the first stage is $0.5$. If the outcome of the first stage is 'No', the probability of the outcome 'Yes' at the second stage is $0.25$.

Calculate the probability of the outcome 'No' then 'Yes'.

Find the probability that the second decision is Yes.

Find the probability that the third decision is Yes.