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

A person chooses 'Yes' or 'No' with probability $0.5$ each. If they choose 'Yes', the probability they choose 'Yes' next is $0.75$; if they choose 'No', the probability they choose 'Yes' next is $0.25$. Find the probability that they choose 'Yes' first and 'No' second.

Under the same probabilities, calculate the probability that both choices are 'Yes'.

Two routes to work are chosen with probabilities $0.6$ (Route 1) and $0.4$ (Route 2). If Route 1 is taken, the probability of arriving on time ('Yes') is $0.9$; if Route 2, it is $0.7$. What is the probability of taking Route 1 and being late ('No')?

A fair coin is flipped. If it lands Heads ('Yes'), then the next 'Yes' probability is $0.6$; if it lands Tails ('No'), the next 'Yes' probability is $0.2$. Find the probability of 'Yes' on both flips.

A student answers two questions. The probability of answering Question 1 correctly ('Yes') is $0.6$. If they answer correctly, the probability of answering Question 2 correctly is $0.8$; if incorrectly, the probability is $0.3$. Find the probability they answer Question 1 correctly and Question 2 incorrectly.

A customer chooses product A with probability $0.7$ and B with probability $0.3$. If A is chosen, the probability of a positive review ('Yes') is $0.8$; if B, it is $0.5$. Find the overall probability of a positive review.

In a three-stage process, the probability of 'Yes' on stage 1 is $0.5$. If Yes on stage 1, stage 2 Yes is $0.8$, otherwise it is $0.3$. If Yes on stage 2, stage 3 Yes is $0.9$, otherwise it is $0.2$. Calculate $P( \text{Yes}_1, \text{No}_2, \text{Yes}_3)$.

An event has three initial outcomes A, B, C with probabilities $0.2$, $0.3$, $0.5$ respectively. After outcome A or B, the probability of success ('Yes') is $0.9$; after C it is $0.4$. Find the overall probability of success.

Using the same three-stage set-up as above, find the probability of exactly one 'Yes' in the three stages.

In the three-stage process of question 7, given that stage 3 is 'Yes', what is the probability that stage 1 was 'No'?