Some Probability Problems

Probability of Series Going to 7 Games

Question: Two teams play a series of games in a best-of-7 series. The first team to win 4 games wins the series. The teams are evenly matched, so each team has a 50% chance of winning each game. What is the probability that the series goes to 7 games?

Solution

To find the probability that the series goes to 7 games, we need to consider the scenario where the first 6 games are split 3-3. Since each team has a 50% chance of winning each game, the probability of this scenario occurring is the probability of getting 3 wins out of 6 games for one team and 3 wins out of 6 games for the other team. The numerator $$\binom{6}{3}$$ represents the number of ways to choose 3 wins out of 6 games. The denominator $$2^6$$ represents the total number of possible outcomes for 6 games. $$ P(\text{3 wins out of 6}) = \frac{\binom{6}{3}}{2^6} $$ $$ P(\text{3 wins out of 6}) = \frac{20}{64} = 0.3125 $$ Therefore, the probability that the series goes to 7 games is $$\boxed{0.3125}$$ or 31.25%.

Bug Reaching a Point in 3D Space

Question: For a bug that starts at (0,0,0). How many ways are there for the bug to get to (4, 4, 4), if the bug can only move up, right, forward.

Solution

There are 12 moves the bug can make to get to (4, 4, 4). There are therefore $$12!$$ ways to order the 12 total moves. The bug must make 4 moves up, 4 moves right, and 4 moves forward. But since each up/right/forward move is indistinguishable, we must divide by $$ 4! $$ for each direction. Therefore, the number of ways the bug can get to (4, 4, 4) is $$ \frac{12!}{4!4!4!} = 34650 $$