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

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.

Using the same two-stage process as above, calculate the probability of the outcome No then Yes.

Again using the same two-stage process, calculate the probability of the outcome Yes then Yes.

In the three-stage conditional decision process of question 5, calculate the probability that there is at least one No among the three decisions.

In the two-stage process above, what is the probability that the second decision is Yes?

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, it is replaced and a white marble is added, making the second-draw probabilities $P(\text{Red})=0.7$, $P(\text{Blue})=0.3$. If a blue marble is drawn first, it is replaced and a green marble is added, making the second-draw probabilities $P(\text{Red})=0.2$, $P(\text{Blue})=0.8$. Calculate the probability of drawing Red then Blue.

In the three-stage process, calculate $P(\text{Yes},\text{Yes},\text{Yes})$.

In the three-stage process, find the probability that the third decision is Yes.

Extend the process to a third stage: after the second decision, if the second is Yes then $P(\text{Yes}\mid\text{Yes}_2)=0.8$ and $P(\text{No}\mid\text{Yes}_2)=0.2$; if the second is No then $P(\text{Yes}\mid\text{No}_2)=0.3$ and $P(\text{No}\mid\text{No}_2)=0.7$. Find $P(\text{Yes at stage 1},\text{No at stage 2},\text{Yes at stage 3})$.

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.

You flip a biased coin three times. The probability of heads on the first flip is $0.5$. If a flip is heads, the next flip has $P(\text{Heads})=0.6$; if a flip is tails, the next has $P(\text{Heads})=0.4$. Find the probability of the sequence Heads, Tails, Heads.

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