Question

a jar contains 8 red marbles numbered 1 to 8 and 12 blue marbles numbered 1 to 12. a marble is drawn at random from the jar. find the probability of the given event. (a) the marble is red your answer is: (b) the marble is odd - numbered your answer is: (c) the marble is red or odd - numbered your answer is: (d) the marble is blue and even - numbered your answer is: question help: video message instructor

Explanation:

Step1: Calculate total number of marbles

The jar has 8 red marbles and 12 blue marbles. So the total number of marbles $n = 8 + 12=20$.

Step2: Calculate probability of red marble (a)

The number of red marbles $n_{red}=8$. The probability $P(\text{red})=\frac{n_{red}}{n}=\frac{8}{20}=\frac{2}{5}$.

Step3: Calculate number of odd - numbered marbles (b)

Among 8 red marbles, 4 are odd - numbered (1, 3, 5, 7). Among 12 blue marbles, 6 are odd - numbered (1, 3, 5, 7, 9, 11). So the number of odd - numbered marbles $n_{odd}=4 + 6 = 10$. The probability $P(\text{odd})=\frac{n_{odd}}{n}=\frac{10}{20}=\frac{1}{2}$.

Step4: Calculate number of red and odd - numbered marbles

The number of red and odd - numbered marbles is 4.

Step5: Calculate probability of red or odd - numbered marbles (c)

Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Here $A$ is the event of getting a red marble and $B$ is the event of getting an odd - numbered marble. $P(A)=\frac{8}{20}$, $P(B)=\frac{10}{20}$ and $P(A\cap B)=\frac{4}{20}$. So $P(A\cup B)=\frac{8 + 10-4}{20}=\frac{14}{20}=\frac{7}{10}$.

Step6: Calculate number of blue and even - numbered marbles (d)

Among 12 blue marbles, 6 are even - numbered (2, 4, 6, 8, 10, 12). So the probability $P(\text{blue and even})=\frac{6}{20}=\frac{3}{10}$.

Answer:

(a) $\frac{2}{5}$
(b) $\frac{1}{2}$
(c) $\frac{7}{10}$
(d) $\frac{3}{10}$