## Teacher Guide

### Probability

#### Getting Started

What is the probability of winning the lottery? It is a question that almost everyone has asked themselves at some point. Thousands of people have devised mathematical formulas that claim to increase your chances of winning. In reality, winning the lottery is just a matter of odds. Explain to your class the probability of winning the Powerball lottery.

Rules: You must choose five correct numbers out of 59 options, including the Powerball number which is one out of 35 options.

- The probability of getting the first ball correct is 5/59 (five numbers chosen /59 possibilities).
- After the first ball, you have reduced the number of possibilities by one, and the number of balls you need to win by one.
- The probability of getting the second ball correct is 4/58.
- The trend continues for the last three balls: 3/57, 2/56, and 1/55.
- The probability of getting the Powerball is separate from the first five balls. The odds are 1/35.
- Multiply all the odds to get your final probability: 5/59 × 4/58 × 3/57 × 2/56 × 1/55 × 1/35 = 1 in 175,223,510.

#### Opening Discussion

Playing card games is another example where probability is important to understand. Games like poker rely heavily on probability since players need to know how likely they are to win or make a hand. It is no coincidence that some of the best poker players in the world are mathematicians. A few basic rules need to be remembered when playing card games:

- Four suits (hearts, spades, diamonds, clubs)
- 13 cards per suit (1-13)
- Two colors (red and black)
- 52 cards total

In the popular card game Texas Hold’em, a player receives two cards to start the game. Explore some of the probabilities of getting a particular hand to start.

- A red Queen and a black King: 2/52 × 2/51 = 0.15%
- A pair of Aces: 4/52 × 3/51 = 0.45%
- Any pair: 52/52 × 3/52 = 5.7%
- A King and Queen of spades: 2/52 × 1/51 = 0.075%
- Any suit King or Queen: 8/52 × 1/51 = 0.30%
- Any two black cards: 26/52 × 25/51 = 24.5%
- Any two diamonds: 13/52 × 12/52 = 5.9%
- Two cards of the same suit: 52/52 × 12/51 = 24%

### Activities:

- Make a Comic
##### Playing the Odds

View Activity - Make a Comic
##### Career Study

View Activity - Extension / Modification
##### Challenge

*(Extension / Modification)*Choose a card game to discuss how using probability is an effective strategy to win.

- Extension / Modification
##### Analyze

*(Extension / Modification)*What is the probability of having three daughters and no sons?

### Concluding Discussion

Discuss with your class how probability can play a negative role in your life. Discuss a variety of surprising probabilities that American youth are facing today. For example:

- 1/3 of black men will be incarcerated
- 3/7 of marriages end in divorce
- 1/4 of women experience domestic violence
- 1/3 of teens have intimate relations by the age of 16
- 1/8 of Americans live in poverty Discuss reasons why these probabilities are so high and ways we can reduce these statistics in the future.