Anne is a perfectly average checkers player. She wins 40 percent of her games, loses 40 percent, and draws the other 20 percent. Assume that these probabilities apply 1 to every game she plays. In the questions below, remember to report probabilities as proportions (on a 0-1 scale) instead of percentages.

3.1. If she plays until she wins a game, what is the probability her first win occurs in her third game?

3.2. If she plays until she wins a game, what is the probability she wins her first game within the first three games she plays?

3.3. If she plays until she wins a game, what is the probability that it takes her more than five games to win her first game?

3.4. What is the expected number of games it will take her to win her first game?

3.5. If she plays until she draws a game, what is the probability that her first draw occurs within the first three games she plays? 3.6. If she plays 10 games, what is the probability that she wins exactly five of them?

3.7. If she plays 10 games, what is the probability that she wins no more than three of them?

3.8. If she plays 10 games, what is the probability that she draws at least two of them?

3.9. Now consider Anne's probability of winning as a variable, instead of being fixed at 40 percent. What would this probability have to be to give her exactly a 50 percent chance of needing no more than three games in order to win her first game?