Question

probability and tree diagrams
use a probability tree to help answer the following questions.
a bag has 2 marbles labeled 2 and 3 and a spinner has 4 options. you randomly pull one marble then spin the spinner once.
what is the probability of:

1. spinning an a and pulling a 2?
2. spinning a consonant or pulling an odd number?
3. spinning a b or a c and not pulling a 2?
4. spinning a letter and pulling a 1?

two spinners each have 3 options. you spin each spinner once.
what is the probability of:

1. spinning a b or spinning a 3?
2. spinning a c or spinning an odd number?
3. not spinning an a and spinning an even number?
4. spinning a vowel and not spinning a 1?

Explanation:

Step1: Calculate probabilities of individual events

The probability of pulling a 2 from the bag with 2 marbles is $P(2)=\frac{1}{2}$. The probability of spinning an 'A' on a spinner with 4 options is $P(A)=\frac{1}{4}$.

Step2: Use multiplication rule for independent - events for question 1

For independent events, the probability of both events occurring is $P(A\cap2)=P(A)\times P(2)=\frac{1}{4}\times\frac{1}{2}=\frac{1}{8}$.

Step3: Identify consonants and odd - numbers and use addition rule for question 2

Consonants on the spinner are B, C, D, so $P(\text{consonant})=\frac{3}{4}$. The odd - number in the bag is 3, so $P(\text{odd})=\frac{1}{2}$. Using the addition rule $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, and since the events are independent $P(\text{consonant}\cap\text{odd})=\frac{3}{4}\times\frac{1}{2}=\frac{3}{8}$. Then $P(\text{consonant}\cup\text{odd})=\frac{3}{4}+\frac{1}{2}-\frac{3}{8}=\frac{6 + 4-3}{8}=\frac{7}{8}$.

Step4: Calculate relevant probabilities for question 3

The probability of spinning a 'B' or 'C' is $P(B\cup C)=\frac{2}{4}=\frac{1}{2}$. The probability of not pulling a 2 (pulling a 3) is $P(\text{not }2)=\frac{1}{2}$. Since they are independent, $P((B\cup C)\cap\text{not }2)=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$.

Step5: Analyze question 4

There is no 1 in the bag, so the probability of pulling a 1 is 0. So the probability of spinning a letter and pulling a 1 is 0.

Step6: For the second set of spinners

The probability of spinning a 'B' on the first spinner is $P(B)=\frac{1}{3}$, the probability of spinning a 3 on the second spinner is $P(3)=\frac{1}{3}$, and $P(B\cap3)=\frac{1}{3}\times\frac{1}{3}=\frac{1}{9}$. Using the addition rule for question 5, $P(B\cup3)=\frac{1}{3}+\frac{1}{3}-\frac{1}{9}=\frac{3 + 3-1}{9}=\frac{5}{9}$.

Step7: Calculate for question 6

The probability of spinning a 'C' on the first spinner is $P(C)=\frac{1}{3}$, the probability of spinning an odd number (1 or 3) on the second spinner is $P(\text{odd})=\frac{2}{3}$, and $P(C\cap\text{odd})=\frac{1}{3}\times\frac{2}{3}=\frac{2}{9}$. Then $P(C\cup\text{odd})=\frac{1}{3}+\frac{2}{3}-\frac{2}{9}=\frac{3 + 2-2}{9}=\frac{7}{9}$.

Step8: Calculate for question 7

The probability of not spinning an 'A' on the first spinner is $P(\text{not }A)=\frac{2}{3}$, the probability of spinning an even number (2) on the second spinner is $P(\text{even})=\frac{1}{3}$, and $P(\text{not }A\cap\text{even})=\frac{2}{3}\times\frac{1}{3}=\frac{2}{9}$.

Step9: Calculate for question 8

The vowel on the first spinner is 'A', $P(A)=\frac{1}{3}$, the probability of not spinning a 1 on the second spinner is $P(\text{not }1)=\frac{2}{3}$, and $P(A\cap\text{not }1)=\frac{1}{3}\times\frac{2}{3}=\frac{2}{9}$.

Answer:

1. $\frac{1}{8}$
2. $\frac{7}{8}$
3. $\frac{1}{4}$
4. 0
5. $\frac{5}{9}$
6. $\frac{7}{9}$
7. $\frac{2}{9}$
8. $\frac{2}{9}$