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The number of $10 bills that a wallet contains is 5.

The number of $5 bills that a wallet contains is 4.

The number of $1 bills that a wallet contains is 6.

From the provided information, the total number of bills is,

\(5 + 4 + 6 = 15\)

Let X represents the number of trials to select the first $10 bill.

The probability of selecting a $10 bill is \(\frac{5}{{15}}\).

The probability that at least two bills must be selected to obtain a first $10 bill is computed as,

\(\begin{aligned}P\left( {X \ge 2} \right) &= 1 - P\left( {X < 2} \right)\\ &= 1 - P\left( {X = 1} \right)\\ &= 1 - \left( {\frac{5}{{15}}} \right){\left( {1 - \frac{5}{{15}}} \right)^{1 - 1}}\\ &= 1 - \frac{1}{3}\\ &= 1 - 0.33\\ &= 0.67\end{aligned}\)

Therefore, the probability that at least two bills must be selected to obtain a first $10 bill is0.67.