Discrete Mathematics
# Conditional Probability

There are two identical boxes in front of you. In box $X,$ there are 3 white balls and 7 black balls. In box $Y,$ there are 7 white balls and 3 black balls. You randomly reach one of the boxes and draw three balls from it one by one. If every time you draw a ball you find it is white, at which of the following stages is the probability the highest that you chose box $X?$

(A) You drew one ball, and it was white.

(B) You drew two balls, and they were both white.

(C) You drew three balls, and they were all white.

Suppose that you are playing 5 card poker. You have 4 cards in your hand and are to receive one more card. If you want to make a full house with the $5^\text{th}$ card, which of the following is the most preferable situation?

(A) You have a three of a kind in your hand.

(B) You have a two pair in your hand.

(C) You have a one pair in your hand.

**Details and Assumptions:**

Do not take other players' cards into consideration.

Suppose that you are playing 5 card poker. You already have 4 cards in your hand and are waiting for the last card. In which of the following situations do you have a higher probability of success?

Situation $A:$ You have a 7 of spade, a 7 of diamond, a 7 of heart, and an ace of club. You want a full house.

Situation $B:$ You have a 6, a 7, a 9, and a 10 of spade. You want a straight.

**Details and Assumptions:**

Do not take the other players' cards into consideration.

Jack Potter has studied how to win a jackpot in Lotto for a long time and recently found out that the numbers 11, 21, 31, 41 (call them "Highs") showed very high winning rates while the numbers 13, 23, 33, 43 (call them "Lows") showed very low winning rates.

Now he is going to buy a Lotto ticket including the Highs in his choice of 6 numbers, because he thinks that their excellent performance in the past must be a good indicator of a higher winning chance in the future.

His brother Harry, however, after listening to Jack's plan, thinks otherwise: Highs showed higher winning rates in the past, so they will show lower winning rates in the future because if all the numbers are equally likely, their winning rates should be averaged out.

Whose reasoning is correct?

**Assumptions and Details:**

Lotto is a form of lottery in which six numbers are drawn from a larger pool (say, 54 or 56). You have to match all six numbers drawn to win the top prize.