Two people receiving gift cards as a gift.

According to an annual survey by Paytronix, around 20% of gift cards are still unused after a full year from purchase. Sounds like free money for businesses, right? It’s all in the accounting - when gift cards are sold, they are counted as a liability until spent. A local restaurant chain sold 100 twenty-five dollar gift cards. Based on the data they expect that 80% of the cards will be used within a year of the purchase. Use the normal approximation along with the continuity correction factor to find the probability that at most 81 gift cards of 100 sold will be used within a year of purchase.
  • Let X\displaystyle {X} be the number of gift cards out of 100 sold that will be used within a year from the purchase. Describe the distribution of X\displaystyle {X} and its parameters:

X∼\displaystyle {X}\sim  ( n=\displaystyle {n}= , p=\displaystyle {p}= )

  • Use the random variable notation to symbolically express the probability that at most 81 gift cards out of 100 sold will be used within a year from the purchase:

  • Let Y\displaystyle {Y} be a normal variable that will be used to approximate the probability in question. Find the parameters of Y\displaystyle {Y} (round the answers to 2 decimal places):

 Y∼\displaystyle {Y}\sim ( μ=\displaystyle \mu= , σ=\displaystyle \sigma= )

  • Use the random variable notation to symbolically express the approximate probability that at most 81 gift cards out of 100 sold will be used within a year from the purchase:

  • Use the correction for continuity:

  • Find the probability (round the answer to 4 decimal places):

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