Question

bob has a coin cup with four $1 tokens and two $5 tokens in it. he also has two $10 tokens and one $25 token in his pocket. he randomly draws a token from the cup, and then randomly draws a token from his pocket. what is the probability that he will draw $30 in tokens? a. \\( \frac { 1 } { 3 } \\) b. \\( \frac { 4 } { 9 } \\) c. \\( \frac { 1 } { 9 } \\) d. \\( \frac { 2 } { 9 } \\)

Explanation:

Step1: Calculate total number of tokens in cup

Total tokens in cup: \(4 + 2=6\)

Step2: Calculate total number of tokens in pocket

Total tokens in pocket: \(2 + 1=3\)

Step3: Calculate combinations for cup draw

Number of ways to draw from cup: \(C(6,1)=\frac{6!}{1!(6 - 1)!}=6\)

Step4: Calculate combinations for pocket draw

Number of ways to draw from pocket: \(C(3,1)=\frac{3!}{1!(3 - 1)!}=3\)

Step5: Calculate total number of combined draws

Total number of combined draws: \(6\times3 = 18\)

Step6: Find favorable cases

To get \(30\), we need \(5\) from cup and \(25\) from pocket.
Number of ways to draw \(5\) from cup: \(C(2,1)=\frac{2!}{1!(2 - 1)!}=2\)
Number of ways to draw \(25\) from pocket: \(C(1,1)=\frac{1!}{1!(1 - 1)!}=1\)
Favorable cases: \(2\times1 = 2\)

Step7: Calculate probability

Probability \(P=\frac{2}{18}=\frac{1}{9}\)

Answer:

C. \(\frac{1}{9}\)