If she is right, the object is worth $25, if she is wrong, the object is worth $4. How high does the probability of the object being authentic have to be for her to take the gamble (meaning purchase the fancy sardine) for $16

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.

9 x 8 divided by (5-3)=36

this may not be at all, but i use the same website you're using for school, and if i'm stuck doing one of these i try to look for it on the section.