Question

a. if he randomly chooses one marble to give to a friend, what is the probability that it is blue? b. tom does not really want to give away blue marbles and would like to change the probability that he chooses a blue marble to 1/10. how many marbles that are not blue could he add to the bag so that the probability of choosing a blue marble becomes 1/10?

Explanation:

1. a. Calculate the probability of choosing a blue - marble (assuming the number of blue marbles and total marbles are known, but since not given in the text, let's assume there are \(b\) blue marbles and \(n\) total marbles):

• The probability \(P\) of an event is given by the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
• In the case of choosing a blue marble, the number of favorable outcomes is the number of blue marbles (\(b\)), and the total number of outcomes is the total number of marbles (\(n\)). So the probability \(P\) of choosing a blue marble is \(P = \frac{b}{n}\).
• Let's assume from the picture - like part of the text where it says "12 - 5", if we assume there are 5 blue marbles and 12 total marbles.
• # Explanation:
• ## Step1: Identify the formula for probability

The formula for probability of an event \(E\) is \(P(E)=\frac{\text{Number of elements in }E}{\text{Total number of elements in the sample - space}}\). Here, the event \(E\) is choosing a blue marble. Let the number of blue marbles be \(b\) and the total number of marbles be \(n\). So \(P=\frac{b}{n}\).

• ## Step2: Substitute the values

Given \(b = 5\) and \(n = 12\), we substitute into the formula: \(P=\frac{5}{12}\).

• # Answer:

\(\frac{5}{12}\)

1. b. Find the number of non - blue marbles to add to change the probability of choosing a blue marble to \(\frac{1}{10}\):

• Let the original number of blue marbles be \(b = 5\) (from part a assumption), and let the number of non - blue marbles to add be \(x\).
• The new total number of marbles is \(n + x=12 + x\), and we want the probability of choosing a blue marble to be \(\frac{1}{10}\).
• Using the probability formula \(P=\frac{b}{n + x}\), and substituting \(P=\frac{1}{10}\) and \(b = 5\), we get \(\frac{5}{12 + x}=\frac{1}{10}\).
• # Explanation:
• ## Step1: Set up the probability equation

We know that the new probability of choosing a blue marble is \(\frac{\text{Number of blue marbles}}{\text{New total number of marbles}}\). Let the number of non - blue marbles to add be \(x\). The number of blue marbles is 5, and the new total number of marbles is \(12 + x\). So \(\frac{5}{12 + x}=\frac{1}{10}\).

• ## Step2: Cross - multiply

Cross - multiplying the equation \(\frac{5}{12 + x}=\frac{1}{10}\) gives us \(5\times10=(12 + x)\times1\), which simplifies to \(50 = 12+x\).

• ## Step3: Solve for \(x\)

Subtract 12 from both sides of the equation \(50 = 12+x\). We get \(x=50 - 12=38\).

• # Answer:

38

Answer:

1. a. Calculate the probability of choosing a blue - marble (assuming the number of blue marbles and total marbles are known, but since not given in the text, let's assume there are \(b\) blue marbles and \(n\) total marbles):

• The probability \(P\) of an event is given by the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
• In the case of choosing a blue marble, the number of favorable outcomes is the number of blue marbles (\(b\)), and the total number of outcomes is the total number of marbles (\(n\)). So the probability \(P\) of choosing a blue marble is \(P = \frac{b}{n}\).
• Let's assume from the picture - like part of the text where it says "12 - 5", if we assume there are 5 blue marbles and 12 total marbles.
• # Explanation:
• ## Step1: Identify the formula for probability

The formula for probability of an event \(E\) is \(P(E)=\frac{\text{Number of elements in }E}{\text{Total number of elements in the sample - space}}\). Here, the event \(E\) is choosing a blue marble. Let the number of blue marbles be \(b\) and the total number of marbles be \(n\). So \(P=\frac{b}{n}\).

• ## Step2: Substitute the values

Given \(b = 5\) and \(n = 12\), we substitute into the formula: \(P=\frac{5}{12}\).

• # Answer:

\(\frac{5}{12}\)

1. b. Find the number of non - blue marbles to add to change the probability of choosing a blue marble to \(\frac{1}{10}\):

• Let the original number of blue marbles be \(b = 5\) (from part a assumption), and let the number of non - blue marbles to add be \(x\).
• The new total number of marbles is \(n + x=12 + x\), and we want the probability of choosing a blue marble to be \(\frac{1}{10}\).
• Using the probability formula \(P=\frac{b}{n + x}\), and substituting \(P=\frac{1}{10}\) and \(b = 5\), we get \(\frac{5}{12 + x}=\frac{1}{10}\).
• # Explanation:
• ## Step1: Set up the probability equation

We know that the new probability of choosing a blue marble is \(\frac{\text{Number of blue marbles}}{\text{New total number of marbles}}\). Let the number of non - blue marbles to add be \(x\). The number of blue marbles is 5, and the new total number of marbles is \(12 + x\). So \(\frac{5}{12 + x}=\frac{1}{10}\).

• ## Step2: Cross - multiply

Cross - multiplying the equation \(\frac{5}{12 + x}=\frac{1}{10}\) gives us \(5\times10=(12 + x)\times1\), which simplifies to \(50 = 12+x\).

• ## Step3: Solve for \(x\)

Subtract 12 from both sides of the equation \(50 = 12+x\). We get \(x=50 - 12=38\).

• # Answer:

38