= Solution
The quoted pass probabilities on later attempts are conditional on reaching those attempts. If $K$ is the first successful attempt, then
$$
P(K=1)=\frac35,\quad P(K=2)=\frac25\frac45=\frac8{25},\quad P(K=3)=\frac25\frac15\frac78=\frac7{100}.
$$
They sum to $99/100$, so conditioning on success by the third attempt is essential. The <conditional mean of a bounded success time> is
$$
\boxed{\mathbb E[K\mid K\leq3]=\frac{1(3/5)+2(8/25)+3(7/100)}{99/100}=\frac{145}{99}\simeq1.465.}
$$
The remaining probability $1/100$ represents failing all three attempts and is excluded by this conditioning.
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