Question

Coin 1 comes up heads with probability \(0.6\) and \(\operatorname{coin} 2\) with probability \(0.5 . \mathrm{A}\) 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 \(\operatorname{coin} 1\) what is the probability that \(\operatorname{coin} 2\) is used on the fifth flip?

Short answer

(a) The proportion of flips that use Coin 1 is \(\frac{5}{9}\). (b) If we start the process with Coin 1, the probability that Coin 2 is used on the fifth flip is 0.0864.

Step-by-step solution

Understand the Problem

We have two coins, Coin 1, which comes up heads with a probability of 0.6, and Coin 2, which comes up heads with a probability of 0.5. We need to find: (a) The proportion of flips that use Coin 1 (b) The probability that Coin 2 is used on the fifth flip if we start with Coin 1

Find the probabilities of obtaining tails

First, let's find the probabilities of obtaining tails for both coins. For Coin 1, the probability of tails is: \(P(T_{1}) = 1 - P(H_{1}) = 1 - 0.6 = 0.4\) For Coin 2, the probability of tails is: \(P(T_{2}) = 1 - P(H_{2}) = 1 - 0.5 = 0.5\)

Find the proportion of flips that use Coin 1

Let \(n_1\) and \(n_2\) be the number of flips for Coin 1 and Coin 2, respectively. The proportion of flips that use Coin 1 is given by: \(\frac{n_1}{n_1+n_2} = \frac{P(T_2)}{P(T_1) + P(T_2)}\) Plugging in the values for \(P(T_1)\) and \(P(T_2)\): \(\frac{n_1}{n_1+n_2} = \frac{0.5}{0.4 + 0.5} = \frac{0.5}{0.9}\) This simplifies to: \(\frac{n_1}{n_1+n_2} = \frac{5}{9}\) So, the proportion of flips that use Coin 1 is \(\frac{5}{9}\).

Find the probability that Coin 2 is used on the fifth flip if we start with Coin 1

For Coin 2 to be used on the fifth flip, Coin 1 must get tails on the fourth flip. This means that Coin 1 must get heads on the first, second, and third flips. The probability of this occurring is: \(P(H_1)^3 * P(T_1) = 0.6^3 * 0.4 = 0.0864\) Thus, the probability that Coin 2 is used on the fifth flip if we start with Coin 1 is 0.0864.

Key concepts

Conditional Probability

Conditional probability is a measure of the likelihood of an event occurring given that another event has already occurred. In the context of our coin flipping problem, if we know that Coin 1 is used first, how does this affect the chances of Coin 2 being used on the fifth flip?

The concept is represented by the notation P(A|B), meaning the probability of event A occurring given that B has already happened. For our case, we were interested in finding the probability that Coin 2 is used on the fifth flip, given that the process started with Coin 1. The solution required us to calculate the probability of obtaining heads three times followed by tails on the fourth flip with Coin 1, which conditioned the use of Coin 2 on the fifth flip.

Using conditional probability allows us to refine our predictions and calculations based on new information, increasing accuracy in probabilistic models.

Probability Models

Probability models are mathematical representations of random events and help in calculating the likelihood of various outcomes. The coin flipping exercise is an example of such a model. We used the probability of tails for each coin to determine the proportion of flips using Coin 1 and the chance of Coin 2 being used on a certain flip.

In creating a probability model, certain assumptions are made - for our coins, we assumed flips are independent and have only two possible outcomes, 'heads' or 'tails', with fixed probabilities. It's critical that the model accurately reflects the scenario being analyzed, as this grounds the validity of the subsequent probabilities calculated from it.

Probability models are fundamental in many disciplines, from games of chance to complex statistical forecasting in finance and meteorology.

Binomial Probability

Binomial probability is a specific type of probability that deals with the number of successes in a fixed number of independent trials, each with the same probability of success. In the context of the exercise, if we consider 'flipping tails' as 'success', then flipping a coin until tails appear can be seen as a binomial experiment.

Binomial probability is calculated using the formula:
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
where \(P(X = k)\) is the probability of k successes in n trials, p is the probability of success on a single trial, and \(\binom{n}{k}\) is a binomial coefficient.

However, in this problem, our task was a bit simpler since we were only looking for the first success (the first tails flipped), rather than a series of successes over multiple flips. Nevertheless, understanding the binomial framework is still key in many probability problems, where the number of trials and successes is a central concern.