A chocolate egg costs $1.20. One chocolate egg in 12, on average, contains a limited edition egg. If a child buys a chocolate egg every week, find the expected total costs of the chocolate eggs bought, up to and including the first egg with a limited edition toy.

A chocolate egg costs $1.20. One chocolate egg in 12, on average, contains a limited edition egg. If a
child buys a chocolate egg every week, find the expected total costs of the chocolate eggs bought, up to
and including the first egg with a limited edition toy.

Answer: the expected total cost of the chocolate eggs bought is $1.20.

Step-by-step explanation:

To find the expected total cost of the chocolate eggs bought, up to and including the first egg with a limited edition toy, we need to consider the probability of getting a limited edition toy in each egg.

The probability of getting a limited edition toy in one chocolate egg is 1/12, as given in the problem.

Let's denote the expected number of chocolate eggs the child needs to buy until they get a limited edition toy as E. We can set up an equation for E using the concept of expected value:

E = 1 + (1/12)(E-1)

Here's how the equation is derived:

The child needs to buy at least one chocolate egg, so we start with 1.

The probability of getting a limited edition toy is 1/12. If the child does not get a limited edition toy in the first egg, they would need to continue buying chocolate eggs until they get one. Therefore, we add (1/12)(E-1) to the expected value.

Simplifying the equation:

E = 1 + (1/12)E - 1/12

12E = 12 + E - 1

11E = 11

E = 1

The expected number of chocolate eggs the child needs to buy until they get a limited edition toy is 1.

Since each chocolate egg costs $1.20, the expected total cost of the chocolate eggs bought, up to and including the first egg with a limited edition toy, is $1.20.

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