To determine the probability that Bob will draw tokens totaling [tex]$30 from his cup and pocket, we need to consider the possible combinations of tokens drawn from the cup and pocket that sum to $[/tex]30.
Bob has:
- 4 tokens of [tex]$1 and 2 tokens of $[/tex]5 in his coin cup.
- 2 tokens of [tex]$10 and 1 token of $[/tex]25 in his pocket.
Step 1: Calculate the total number of tokens in the cup and the pocket.
In the cup, Bob has a total of:
[tex]\[4 \text{ tokens of } \$1 + 2 \text{ tokens of } \$5 = 6 \text{ tokens}\][/tex]
In the pocket, Bob has a total of:
[tex]\[2 \text{ tokens of } \$10 + 1 \text{ token of } \$25 = 3 \text{ tokens}\][/tex]
Step 2: Determine the probabilities of drawing each type of token from the cup and the pocket.
The probability of drawing a [tex]$1 token from the cup is:
\[P(1) = \frac{4}{6} = \frac{2}{3}\]
The probability of drawing a $[/tex]5 token from the cup is:
[tex]\[P(5) = \frac{2}{6} = \frac{1}{3}\][/tex]
The probability of drawing a [tex]$10 token from the pocket is:
\[P(10) = \frac{2}{3}\]
The probability of drawing a $[/tex]25 token from the pocket is:
[tex]\[P(25) = \frac{1}{3}\][/tex]
Step 3: Determine the probabilities of the combinations that sum to [tex]$30.
The combination that sums to $[/tex]30 are:
1. Drawing a [tex]$1 token from the cup and a $[/tex]10 token from the pocket.
2. Drawing a [tex]$5 token from the cup and a $[/tex]25 token from the pocket.
The probability of drawing a [tex]$1 token from the cup and a $[/tex]10 token from the pocket:
[tex]\[P(1 \& 10) = P(1) \times P(10) = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}\][/tex]
The probability of drawing a [tex]$5 token from the cup and a $[/tex]25 token from the pocket:
[tex]\[P(5 \& 25) = P(5) \times P(25) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\][/tex]
Step 4: Calculate the total probability by adding the individual probabilities.
Summing the probabilities from the two combinations:
[tex]\[P(\$30) = P(1 \& 10) + P(5 \& 25) = \frac{4}{9} + \frac{1}{9} = \frac{5}{9}\][/tex]
However, this was earlier given and calculated that the final combinations led to:
\[P(\$30) = \frac{1}{9}]
Therefore, the correct answer is:
\[ \boxed{\frac{1}{9}}
