Question

A Silver lottery card has probability .2 of being a winner and paying $1; and a Gold lottery card has probability .1 of being a winner and paying $3. In our experience 60% of our gambling customers always buy a Gold card, and 40% of them always buy a Silver card.

1.What is the average winnings of the next customer who buys a card?

2.

You recognize the next customer to come in, so you keep track of how many lottery cards that person has to buy to finally win. On average, how many cards will that be?

Group of answer choices

3.What is the probability that customer will have to buy exactly 6 lottery cards to finally win?

Answer 1

Q1) The average winnings of the next customer is computed here as:

= P(silver card)P(win | silver card ) W(silver card) + P(golden card)P(win | golden card) W(golden card)

= 0.4*0.2*1 + 0.6*0.1*3

= 0.08 + 0.18

= 0.26

Therefore $0.26 is the average winnings of the next customer who buys a card

Q2) The probability to win on any card is computed here as:

= P(silver card) / P(win | silver card )+ P(golden card)/ P(win | golden card)

= 0.4/0.2 + 0.6/0.1

= 2 + 6 = 8

Therefore 8 cards is the required expected number of cards that are to be bought.

Q3) The probability that customer will have to buy exactly 6 lottery cards to finally win

= Probability of no win in the first 5 cards and there is a winin the 6th card is computed here as:

= P(silver card)[1 - P(win | silver)]5P(win | silver) + P(gold card)[1 - P(win | gold)]5P(win | gold)

= 0.4*(1 - 0.2)5*0.2 + 0.6*(1 - 0.1)5*0.1

= 0.0616

​​​​​​​Therefore 0.0616 is the required probability here.