Analysis of Coin Flipping in a Two-Coin System

Coin 1 comes up heads with probability 0.6 and coin 2 with probability 0.5. A coin

is continually flipped until it comes up tails, at which time that coin is put aside and

we start flipping the other one.

(a) What proportion of flips use coin 1?

(b) If we start the process with coin 1 what is the probability that coin 2 is used

on the fifth flip?

In this blog post, we will be solving a problem involving two coins - Coin 1 and Coin 2. The probability of Coin 1 coming up heads is 0.6, while the probability of Coin 2 coming up heads is 0.5. The problem is to find the proportion of flips that use Coin 1 and the probability that Coin 2 is used on the fifth flip when we start with Coin 1.

A coin is continually flipped until it comes up tails, at which time that coin is put aside, and we start flipping the other coin. This can be modeled as a Markov Chain, with the state being either Coin 1 or Coin 2.

(a) To find the proportion of flips that use Coin 1, we need to find the expected number of flips for each coin. Let's define N1 as the expected number of flips for Coin 1, and N2 as the expected number of flips for Coin 2. We know that N1 = 1/0.6 + N2 and N2 = 1/0.5 + N1. Solving these equations, we get N1 = 2.3333 and N2 = 2. Hence, the proportion of flips using Coin 1 is N1 / (N1 + N2) = 2.3333 / (2.3333 + 2) = 0.5417.

(b) If we start with Coin 1, the probability that Coin 2 is used on the fifth flip can be calculated using the Markov Chain transition matrix. Let's define Pij as the probability of being in state j after i flips, given that we start in state i. The transition matrix is given by

P = [(0.6, 0.4), (0.5, 0.5)]

The probability that Coin 2 is used on the fifth flip can be calculated as P12^4 * 0.5, where P12 is the probability of transitioning from state 1 (Coin 1) to state 2 (Coin 2). Hence, the answer is (0.4)^4 * 0.5 = 0.0256.

In conclusion, we have solved a problem involving two coins and found the proportion of flips that use Coin 1 and the probability that Coin 2 is used on the fifth flip when we start with Coin 1. This problem can be modeled as a Markov Chain and solved using mathematical analysis.