Chapter 11.7, Problem 7C

To determine

To calculate: The probability of drawing a red marble and a blue marble.

Answer to Problem 7C

The probability of drawing a red and a blue marble from a bag of $\text{12}$ different colored marbles is $\frac{1}{66}$

Explanation of Solution

Given information:

Consider the information provided in the question:

There are marbles $\text{12}$ in a bag. Each marble is a different color, including red and blue.

Two marbles are drawn at random.

Formula used:

To find the number of combinations of $n$ objects taken $r$ at a time, divide the number of permutations of $n$ objects taken $r$ at a time by $r!$

  ${}_n{C}_r=\frac{{}_n{P}_r}{r!}$

Calculation:

As per the problem,

Total number of marbles in the bag = $\text{12}$

Number of marbles drawn at random = $\text{2}$

Number of combinations in which two marbles can be drawn:

  ${}_{12}{C}_2=\frac{{}_{12}{P}_2}{2!}$ [Use combinations formula: ${}_n{C}_r=\frac{{}_n{P}_r}{r!}$ ]

  $=\frac{12\cdot 11}{2\cdot 1}$ [Write ${}_{12}{P}_2$ and $2!$ as products]

  $=66$ [Divide out common factors and simplify]

There are $66$ different combinations of marbles that can be drawn. Only one of the combinations includes a red and a blue marble.

So, the probability is $\frac{1}{66}$ .

Hence,

The probability of drawing a red and a blue marble from a bag of $\text{12}$ different colored marbles is $\frac{1}{66}$