Question

a jar contains 28 red marbles numbered 1 to 28 and 30 blue marbles numbered 1 to 30. a marble is drawn at random from the jar. find the probability of the given event. please enter 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: Find total number of marbles

Total marbles = red marbles + blue marbles = \(28 + 30 = 58\)

Step2: Part (a) - Probability the marble is red

Number of red marbles = \(28\)
Probability \(P(\text{red})=\frac{\text{Number of red marbles}}{\text{Total number of marbles}}=\frac{28}{58}=\frac{14}{29}\)

Step3: Part (b) - Probability the marble is odd - numbered

• For red marbles (numbered 1 - 28): Number of odd - numbered red marbles. The number of odd numbers from 1 to 28 is \(\frac{28}{2}=14\) (since between 1 - n, if n is even, number of odd numbers is \(\frac{n}{2}\)).
• For blue marbles (numbered 1 - 30): Number of odd - numbered blue marbles. The number of odd numbers from 1 to 30 is \(\frac{30}{2}=15\) (since between 1 - n, if n is even, number of odd numbers is \(\frac{n}{2}\)).

Total number of odd - numbered marbles = \(14 + 15=29\)
Probability \(P(\text{odd})=\frac{\text{Number of odd - numbered marbles}}{\text{Total number of marbles}}=\frac{29}{58}=\frac{1}{2}\)

Step4: Part (c) - Probability the marble is red or odd - numbered

We use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), where \(A\) is the event that the marble is red and \(B\) is the event that the marble is odd - numbered.

• \(P(A) = \frac{28}{58}\) (from part a)
• \(P(B)=\frac{29}{58}\) (from part b)
• \(P(A\cap B)\): Number of marbles that are red and odd - numbered. As we found, number of odd - numbered red marbles is 14. So \(P(A\cap B)=\frac{14}{58}\)

\(P(\text{red or odd})=\frac{28}{58}+\frac{29}{58}-\frac{14}{58}=\frac{28 + 29-14}{58}=\frac{43}{58}\)

Step5: Part (d) - Probability the marble is blue or even - numbered

First, find the number of blue marbles (\(n(\text{blue}) = 30\)), number of even - numbered marbles and \(n(\text{blue}\cap\text{even})\)

• Number of even - numbered red marbles: From 1 - 28, number of even numbers is 14.
• Number of even - numbered blue marbles: From 1 - 30, number of even numbers is 15.
• Total number of even - numbered marbles \(n(\text{even})=14 + 15 = 29\)
• Number of marbles that are blue and even - numbered: Number of even - numbered blue marbles is 15. So \(n(\text{blue}\cap\text{even}) = 15\)

Using the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), where \(A\) is the event that the marble is blue and \(B\) is the event that the marble is even - numbered.
\(P(\text{blue})=\frac{30}{58}\), \(P(\text{even})=\frac{29}{58}\), \(P(\text{blue}\cap\text{even})=\frac{15}{58}\)
\(P(\text{blue or even})=\frac{30}{58}+\frac{29}{58}-\frac{15}{58}=\frac{30 + 29-15}{58}=\frac{44}{58}=\frac{22}{29}\)

Answer:

s:
(a) \(\frac{14}{29}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{43}{58}\)
(d) \(\frac{22}{29}\)