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

Question 1

Skill question

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

Find the overall probability of success.

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Question 2

Skill question

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.

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Question 3

Skill question

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.

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Question 4

Skill question

An individual makes two consecutive choices. The probability that the first choice is 'Yes' is $0.5$ . If the first choice is 'Yes', the probability that the second choice is 'Yes' is $0.75$ .

Calculate the probability that both choices are 'Yes'.

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Question 5

Skill question

The question tests the ability to calculate the probability of the intersection of two events using the conditional probability formula: $P (A \cap B) = P (A) P (B ∣ A)$ .

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 is taken, the probability of arriving on time is $0.7$ . What is the probability of taking Route 1 and being late ('No')?

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Question 6

Skill question

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 that the coin lands 'Yes' on both flips.

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Question 7

Skill question

A process consists of three stages. At each stage, the result can be either 'Yes' or 'No'. The probabilities for each stage depend on the result of the previous stage, as shown in the tree diagram below:

Find the probability that there is exactly one 'Yes' across the three stages.

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Question 8

Skill question

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.

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Question 9

Skill question

Consider a three-stage process where each stage results in either 'Yes' or 'No'. The probabilities are defined as follows:

The probability of 'Yes' at Stage 1 is $P (Yes_{1}) = 0.5$ .
The probability of 'Yes' at Stage 2 depends on the outcome of Stage 1:
- $P (Yes_{2} ∣ Yes_{1}) = 0.8$
- $P (Yes_{2} ∣ No_{1}) = 0.3$
The probability of 'Yes' at Stage 3 depends on the outcome of Stage 2:
- $P (Yes_{3} ∣ Yes_{2}) = 0.9$
- $P (Yes_{3} ∣ No_{2}) = 0.2$

Given that the outcome of stage 3 is 'Yes', calculate the probability that the outcome of stage 1 was 'No'.

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Question 10

Skill question

In a three-stage process, the probability of obtaining 'Yes' on stage 1 is $0.5$ . If the result of stage 1 is 'Yes', the probability of 'Yes' on stage 2 is $0.8$ , otherwise it is $0.3$ . If the result of stage 2 is 'Yes', the probability of 'Yes' on stage 3 is $0.9$ , otherwise it is $0.2$ .

Calculate $P (Yes_{1} \cap No_{2} \cap Yes_{3})$ .

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IB Mathematics Applications & Interpretation Question Type 3: Given a Worded Problem, Converting to a Tree Diagram to Calculate Probability of an Outcome Exercises