## Return to Fred's Fun Factory (with 50 cents)

In order to play a popular “spinning wheel” game at Fred's Fun Factory Arcade, a player is required to pay a small, fixed amount of 25 cents each time he/she wants to make the wheel spin. When the wheel stops, the player is awarded tickets based on where the wheel stops -- and these tickets are then redeemable for prizes at a redemption center within the arcade.

This particular game has no skill component; each spin of the wheel is a random event, and the results from each spin of the wheel are independent of the results of previous spins.

The wheel awards tickets with the following probabilities:

1 ticket | 35% |

2 tickets | 20% |

3 tickets | 20% |

5 tickets | 10% |

10 tickets | 10% |

25 tickets | 4% |

100 tickets | 1% |

A young girl is given 2 quarters so that she can play the game two times. Let $X$ be the number of tickets she wins based on two spins. There are 26 possible values for $X$ that the young girl can obtain in this case, and those values are listed to the right.

Some values of $X$ are more common than others. For example, winning only 2 tickets in two spins is a somewhat common occurrence with probability 0.1225 as it means the player earns 1 ticket on the first spin and 1 ticket on the second spin. Similarly, winning 200 tickets in two spins is a somewhat rare occurrence with probability 0.0001 as it means the player earns 100 tickets on the first spin and 100 tickets on the second spin. A full list of the possible values of $X$ and the corresponding probabilities for almost every value of $X$ is shown at right.

- Four probability values are deliberately hidden. Determine the 4 missing probability values in the distribution. (Hint: since all values of $X$ are listed, and since the probabilities that are shown add up to 0.66, the 4 hidden probabilities you are computing should add up to very specific value.)
- Which value of $X$ is most common?
- The young girl considers it a "good day" with the game if she wins more than 100 tickets based on 2 spins. What is the probability that she will have a "good day" based on that definition?