Answer:
The expected value for an event with outcomes:
{x₁, x₂, ..., xₙ}
Each one with probability:
{p₁, p₂, ..., pₙ}
Is just:
Ev = x₁*p₁ + ... + xₙ*pₙ
here we have two outcomes:
x₁ = the object worths $25
x₂ = the object is worth $4.
Each one with probability p₁ and p₂ respectively, such that:
p₁ + p₂ = 1
Then the expected value is:
Ev = p₁*($25) + p₂*($4)
Now we want to know how should be the probabilities, such that buying the object for $16 is whort.
Well, the purchase will be whort if the expected value is larger than $16.
This is equivalent to:
p₁*($25) + p₂*($4) - $16 > $0
Knowing that:
p₁ + p₂ = 1
we can rewrite:
p₂ = 1 - p₁
replacing that in the above inequality we get:
p₁*($25) + ( 1 - p₁)*($4) - $16 > $0
Now we can solve this for p₁
p₁*($25 - $4) + $4 - $16 > $0
p₁*$21 - $12 > $0
p₁*$21 > $12
p₁ > $12/$21 = 0.571
The probability of the object being authentic should be larger than 0.571 to take the gamble.
