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 probability of pulling a 2 and spinning an 'A'

The probability of pulling a 2 from 2 marbles is $\frac{1}{2}$, and the probability of spinning an 'A' from 4 options is $\frac{1}{4}$. Since these are independent events, we multiply the probabilities. So the probability is $\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}$.

Step2: Calculate probability of spinning a consonant or pulling an odd number

The probability of pulling an odd - numbered marble (the number 3) is $\frac{1}{2}$. The consonants among A, B, C, D are B, C, D, so the probability of spinning a consonant is $\frac{3}{4}$. Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, $P(A)=\frac{1}{2}$, $P(B)=\frac{3}{4}$, and $P(A\cap B)=\frac{1}{2}\times\frac{3}{4}=\frac{3}{8}$. Then $P(A\cup B)=\frac{1}{2}+\frac{3}{4}-\frac{3}{8}=\frac{4 + 6-3}{8}=\frac{7}{8}$.

Step3: Calculate probability of spinning a 'B' or 'C' and not pulling a 2

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

Step4: Calculate probability of spinning a letter and pulling a 1

There are no 1 - labeled marbles, so the probability is 0.

Step5: Calculate probability of spinning a 'B' or spinning a 3

For the first spinner with 3 options (A, B, C), $P(B)=\frac{1}{3}$. For the second spinner with 3 options (1, 2, 3), $P(3)=\frac{1}{3}$, and $P(B\cap3)=\frac{1}{3}\times\frac{1}{3}=\frac{1}{9}$. Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, we have $P(A\cup B)=\frac{1}{3}+\frac{1}{3}-\frac{1}{9}=\frac{3 + 3-1}{9}=\frac{5}{9}$.

Step6: Calculate probability of spinning a 'C' or spinning an odd number

For the first spinner, $P(C)=\frac{1}{3}$. For the second spinner, odd numbers are 1 and 3, so $P(\text{odd})=\frac{2}{3}$, and $P(C\cap\text{odd})=\frac{1}{3}\times\frac{2}{3}=\frac{2}{9}$. Then $P(C\cup\text{odd})=\frac{1}{3}+\frac{2}{3}-\frac{2}{9}=\frac{3 + 6 - 2}{9}=\frac{7}{9}$.

Step7: Calculate probability of not spinning an 'A' and spinning an even number

The probability of not spinning an 'A' on the first spinner is $\frac{2}{3}$. The probability of spinning an even number (2) on the second spinner is $\frac{1}{3}$. Since they are independent events, the probability is $\frac{2}{3}\times\frac{1}{3}=\frac{2}{9}$.

Step8: Calculate probability of spinning a vowel and not spinning a 1

The vowel on the first spinner is A, so $P(\text{vowel})=\frac{1}{3}$. The probability of not spinning a 1 on the second spinner is $\frac{2}{3}$. Since they are independent events, the probability is $\frac{1}{3}\times\frac{2}{3}=\frac{2}{9}$.

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. $\frac{2}{9}$