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Answer :
Answer:
The expected number of keys she will try before she opens the door is 4.5.
Step-by-step explanation:
Probability of opening with the first key:
One key out of 8 working, so: 1/8 probability of opening with the first key, that is:
[tex]P(X = 1) = \frac{1}{8}[/tex]
Probability of opening with the second key:
Doesn't work on the first(7/8 probability), works on the second(1/7 probability). So
[tex]P(X = 2) = \frac{7}{8}*\frac{1}{7} = \frac{1}{8}[/tex]
Following the logic:
For each value of x from 1 to 8, we have that [tex]P(X = x) = \frac{1}{8}[/tex]
Find the expected number of keys she will try before she opens the door.
Each outcome multiplied by its probability. So
[tex]E(X) = 1\frac{1}{8} + 2\frac{1}{8} + 3\frac{1}{8} + 4\frac{1}{8} + 5\frac{1}{8} + 6\frac{1}{8} + 7\frac{1}{8} + 8\frac{1}{8} = \frac{1+2+3+4+5+6+7+8}{8} = 4.5[/tex]
The expected number of keys she will try before she opens the door is 4.5.
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