July 8 - Infinite Probability

July 8 - Infinite Probability

Roman Hall

Infinite Probability

A relatively common thing in probability is when there are infinitely many outcomes and the probabilities of those outcomes actually form a geometric series that we can sum up. Let's look at a few examples.

Open Ended

1981 AHSME #26 - Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six.

Solution to 1981 AHSME Problem 26

https://artofproblemsolving.com/wiki/index.php/1981_AHSME_Problems/Problem_26

Open Ended

2021 AKML #18 - Brynn and Finley are playing a game in which they alternately roll a standard six-sided die. The loser of the game is the first person to roll the same number that the other person just rolled. For example, if Brynn rolls a 1, then Finley will lose if she rolls a 1 on her next roll. Brynn rolls first, and the game continues until someone loses. Compute the probability that Brynn will lose. Express your answer as a common fraction.

Solution to 2021 AKML #18

With each roll, the person rolling will lose with probability  $\frac{1}{6}$ , and the game will continue with probability  $\frac{5}{6}$ . Brynn cannot lose on her first roll, but she can lose on her second roll with probability  $\frac{5}{6}\left(\frac{1}{6}\right)$ , on her third roll with probability  $\left(\frac{5}{6}\right)^3\left(\frac{1}{6}\right)$ , on her fourth roll with probability  $\left(\frac{5}{6}\right)^5\left(\frac{1}{6}\right)$  and so on. Summing that up as an infinite geometric series gives the requested probability of  $\frac{5}{11}$ .

Open Ended

2021 ARML #5 - A deck of cards consists of 2 purple cards and 6 orange cards. Zak draws a hand of three cards at random from the deck. If all three cards are orange, he returns them to the deck, shuffles the deck, and repeats the entire process until he has a hand containing at least one purple card. Then he draws one additional card. Compute the probability that this last card is purple.

Solution to 2021 ARML #5

Send me the word "penguin" if you want the solution. Even if you don't want the solution, it'd be a good idea to send it to me anyway