Question

section 5.2 homework score: 9.67/15 answered: 10/15 question 11 a jar contains 12 red marbles numbered 1 to 12 and 8 blue marbles numbered 1 to 8. a marble is drawn at random from the jar. find the probability of the given event, please show your answers as reduced fractions. (a) the marble is red. p(red)= (b) the marble is odd - numbered. p(odd)= (c) the marble is red or odd - numbered. p(red or odd) = (d) the marble is blue or even - numbered. p(blue or even) =

Explanation:

Step1: Calculate total marbles

The total number of marbles is \(12 + 8=20\).

Step2: Calculate P(red)

There are 12 red marbles. So \(P(\text{red})=\frac{12}{20}=\frac{3}{5}\).

Step3: Calculate number of odd - numbered marbles

There are 6 odd - numbered red marbles (1, 3, 5, 7, 9, 11) and 4 odd - numbered blue marbles (1, 3, 5, 7). So there are \(6 + 4 = 10\) odd - numbered marbles. Then \(P(\text{odd})=\frac{10}{20}=\frac{1}{2}\).

Step4: Calculate P(red and odd)

There are 6 odd - numbered red marbles. So \(P(\text{red and odd})=\frac{6}{20}=\frac{3}{10}\).

Step5: Calculate P(red or odd)

Using the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), we have \(P(\text{red or odd})=P(\text{red})+P(\text{odd})-P(\text{red and odd})=\frac{3}{5}+\frac{1}{2}-\frac{3}{10}=\frac{6 + 5- 3}{10}=\frac{8}{10}=\frac{4}{5}\).

Step6: Calculate P(blue)

There are 8 blue marbles. So \(P(\text{blue})=\frac{8}{20}=\frac{2}{5}\).

Step7: Calculate P(even)

There are \(12 - 6=6\) even - numbered red marbles and \(8 - 4 = 4\) even - numbered blue marbles. So there are \(6+4 = 10\) even - numbered marbles. Then \(P(\text{even})=\frac{10}{20}=\frac{1}{2}\).

Step8: Calculate P(blue and even)

There are 4 even - numbered blue marbles. So \(P(\text{blue and even})=\frac{4}{20}=\frac{1}{5}\).

Step9: Calculate P(blue or even)

Using the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), we have \(P(\text{blue or even})=P(\text{blue})+P(\text{even})-P(\text{blue and even})=\frac{2}{5}+\frac{1}{2}-\frac{1}{5}=\frac{4 + 5- 2}{10}=\frac{7}{10}\).

Answer:

(a) \(\frac{3}{5}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{4}{5}\)
(d) \(\frac{7}{10}\)