A grocery store is running a sales promotion where a customer will receive one o

Question

A grocery store is running a sales promotion where a customer will receive one of the letters A, E, L, S, U and V for each purchase. The letters are given away randomly by a cashier. If a customer collects all six letters (“VALUES”), he/she will get a coupon for $10. What is the expected number of purchases needed to get a coupon. (Hint: Let X be the number of purchases needed to collect all six letters. Let Xi be the number of purchases to get the ith missing letter, for i = 1, . . . , 6, i.e., X = X1 + X2 + · · · + X6. To find the corresponding probabilities, notice that the shopper always needs to get a letter he does not have yet. For X1, any letter is ‘good’, for X2, only 5 out of 6 letters will do, for X3, only 4 out of 6 will give him what he needs to complete VALUES, etc.)

Explanation / Answer

X : Number of purchases needed to get all the alphabets

X = X1 + X2 + ... + X6

Xi : Number of purchases needed to get the i th alphabet

The first letter can be any of the 6 letters.

The second letter can be one of the remaining 5 letters out of 6.

The third letter can be one of the 4 letters out of 6.

The fourth letter can be one of the 3 out of 6 and so on.

P( X) = (6/6)(5/6)(4/6)(3/6)(2/6)(1/6) = 5/324 = 0.0154

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