Question

probability
solve the following using any method.
a bag has 4 red and 3 green marbles, and a regular die has 6 sides. you randomly pull 1 marble from the bag and roll the die once.
what is the probability of:

1. pulling a green marble and rolling a number less than 4?
2. pulling a red marble and not rolling a 3?
3. pulling a blue marble and rolling a number greater than 5?

there are four card suits and a spinner has 3 options. you draw one card and spin the spinner once.
what is the probability of:

1. spinning an even number and drawing a heart?
2. spinning an odd number and not drawing a heart?
3. drawing a club or spade and spinning the number 3?
4. drawing a diamond and spinning a prime number?

Explanation:

Step1: Recall probability formula

The probability of an event $A$ is $P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. For two independent events $A$ and $B$, the probability of both $A$ and $B$ occurring is $P(A\cap B)=P(A)\times P(B)$.

Step2: Calculate probabilities for part 1

• Probability of pulling a green marble: There are $4$ red and $3$ green marbles, so the total number of marbles is $4 + 3=7$. The probability of pulling a green marble, $P(G)=\frac{3}{7}$.
• Probability of rolling a number less than $4$ on a die: The numbers less than $4$ on a die are $1$, $2$, and $3$. So $P(<4)=\frac{3}{6}=\frac{1}{2}$.
• Then $P(G\cap<4)=\frac{3}{7}\times\frac{1}{2}=\frac{3}{14}$.

Step3: Calculate probabilities for part 2

• Probability of pulling a red marble: $P(R)=\frac{4}{7}$.
• Probability of not rolling a $3$ on a die: The probability of rolling a $3$ is $\frac{1}{6}$, so the probability of not rolling a $3$ is $P(\text{not }3)=1 - \frac{1}{6}=\frac{5}{6}$.
• Then $P(R\cap\text{not }3)=\frac{4}{7}\times\frac{5}{6}=\frac{20}{42}=\frac{10}{21}$.

Step4: Calculate probabilities for part 3

• Probability of pulling a blue marble: There are no blue marbles, so $P(B) = 0$.
• Probability of rolling a number greater than $5$ on a die: The number greater than $5$ on a die is $6$, so $P(>5)=\frac{1}{6}$.
• Then $P(B\cap>5)=0\times\frac{1}{6}=0$.

Step5: Calculate probabilities for part 4

• Probability of spinning an even number on a spinner with $3$ options ($1$, $2$, $3$): The even number is $2$, so $P(\text{even})=\frac{1}{3}$.
• Probability of drawing a heart from $4$ card - suits: $P(\text{heart})=\frac{1}{4}$.
• Then $P(\text{even}\cap\text{heart})=\frac{1}{3}\times\frac{1}{4}=\frac{1}{12}$.

Step6: Calculate probabilities for part 5

• Probability of spinning an odd number on a spinner: The odd numbers are $1$ and $3$, so $P(\text{odd})=\frac{2}{3}$.
• Probability of not drawing a heart: The probability of drawing a heart is $\frac{1}{4}$, so the probability of not drawing a heart is $P(\text{not heart}) = 1-\frac{1}{4}=\frac{3}{4}$.
• Then $P(\text{odd}\cap\text{not heart})=\frac{2}{3}\times\frac{3}{4}=\frac{1}{2}$.

Step7: Calculate probabilities for part 6

• Probability of drawing a club or spade: There are $2$ suits out of $4$ that are club or spade, so $P(\text{club or spade})=\frac{2}{4}=\frac{1}{2}$.
• Probability of spinning the number $3$ on a spinner: $P(3)=\frac{1}{3}$.
• Then $P((\text{club or spade})\cap3)=\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}$.

Step8: Calculate probabilities for part 7

• Probability of drawing a diamond: $P(\text{diamond})=\frac{1}{4}$.
• Probability of spinning a prime number on a spinner: The prime numbers on the spinner ($1$, $2$, $3$) are $2$ and $3$, so $P(\text{prime})=\frac{2}{3}$.
• Then $P(\text{diamond}\cap\text{prime})=\frac{1}{4}\times\frac{2}{3}=\frac{1}{6}$.

Answer:

1. $\frac{3}{14}$
2. $\frac{10}{21}$
3. $0$
4. $\frac{1}{12}$
5. $\frac{1}{2}$
6. $\frac{1}{6}$
7. $\frac{1}{6}$