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  4. A real estate man has eight master keys to open several new houses. Only one master key will open a given house. If 40% of these homes are usually left unlocked, the probability that the real estate man can get into a specific home if he selects three master keys at random, is

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Q. A real estate man has eight master keys to open several new houses. Only one master key will open a given house. If 40% of these homes are usually left unlocked, the probability that the real estate man can get into a specific home if he selects three master keys at random, is

Probability - Part 2

Solution:

U : Home is unlocked: $P(U) = 2/5 $
L : Home is locked; $P(L) = 3/5$
A : room is opened by any of the 3 keys
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$P ( A )= P ( A \cap U )+ P ( A \cap L )= P ( U ) \cdot P ( A / U )+ P ( L ) \cdot P ( A / L ) $
$P ( A )=\frac{2}{5} \cdot 1+\frac{3}{5} \cdot P ( A / L ) $
$P ( A / L )=\frac{{ }^7 C _2}{{ }^8 C _3}=\frac{21}{56}=\frac{3}{8} \left[ k _1 k _2 k _3 k _4 k _5 k _6 k _7 k _8\right]$
$\therefore P ( A )=\frac{2}{5}+\frac{3}{5} \cdot \frac{3}{8}=\frac{2}{5}+\frac{9}{40}=\frac{16+9}{40}=\frac{5}{8} $