Problem 70 - A First Course in Probability

Question

Consider an urn containing a large number of coins, and suppose that each of the coins has some probability \(p\) of turning up heads when it is flipped. However, this value of \(p\) varies from coin to coin. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the \(p\) -value of the coin can be regarded as being the value of a random variable that is uniformly distributed over \([0,1] .\) If a coin is selected at random from the urn and flipped twice, compute the probability that (a) the first flip results in a head; (b) both flips result in heads.

Step-by-Step Solution

Answer

(a) The probability that the first flip results in a head is \(\frac{1}{2}\). (b) The probability that both flips result in heads is \(\frac{1}{3}\).

Step 1: (a) Probability of first flip resulting in a head

We need to find the probability that the first flip of the coin results in a head. This probability can be represented as: \(P(\text{First flip is a head}) = \int_{0}^{1}pP(p)dp\) Since \(P(p) = 1\) for \(0 \leq p \leq 1\): \(P(\text{First flip is a head}) = \int_{0}^{1}pdp\) Now, integrating with respect to p: \(P(\text{First flip is a head}) = \frac{1}{2}p^2\Big|_0^1\) \(P(\text{First flip is a head}) = \frac{1}{2}\) So, the probability that the first flip results in a head is \(\frac{1}{2}\).

Step 2: (b) Probability of both flips resulting in heads

Now, we need to find the probability that both flips result in heads. This probability can be represented as: \(P(\text{Both flips are heads}) = \int_{0}^{1}p^2P(p)dp\) Again, since \(P(p) = 1\) for \(0 \leq p \leq 1\): \(P(\text{Both flips are heads}) = \int_{0}^{1}p^2dp\) Now, integrating with respect to p: \(P(\text{Both flips are heads}) = \frac{1}{3}p^3\Big|_0^1\) \(P(\text{Both flips are heads}) = \frac{1}{3}\) So, the probability that both flips result in heads is \(\frac{1}{3}\). In summary, the probabilities found are: - Probability of first flip resulting in a head: \(\frac{1}{2}\) - Probability of both flips resulting in heads: \(\frac{1}{3}\)

Key Concepts

Uniform Distribution

When we talk about the uniform distribution, what we're referring to is a type of probability distribution where all outcomes are equally likely. Imagine laying a flat, level ruler on a table; every inch of the ruler has the same chance to be picked if you were to randomly select a point. This is how uniform distribution behaves; but in our case, it's not inches on a ruler but the probability, or value of ‘p’, of flipping a head on any given coin.

Uniform distribution is characterized by two parameters: the minimum and maximum values, usually denoted as ‘a’ and ‘b’, respectively. In our urn example with the coins, this ‘a’ and ‘b’ are from the range 0 to 1. Thus, every probability value between 0 and 1 is equally likely for each coin selected, symbolically this can be expressed as a continuous uniform distribution over the interval \[0, 1\].

One of the most noteworthy properties of the uniform distribution is that its probability density function is constant. Since each outcome within a prescribed range is equally likely, the uniform distribution is a go-to model for representing fairness in a set of outcomes, such as the chance of any particular coin being picked from the urn.

Probability Density Function

The Probability Density Function, or PDF, is a function that describes the likelihood of a random variable taking on a particular value. To speak simply, it's like a bell curve for grades in a class; it shows where the most common grades are concentrated. However, for the uniform distribution, the PDF is more like a flat line since all outcomes are equally probable, representing that each student has an equal chance to get any grade from the lowest to the highest.

Essentially, the PDF helps determine the chances of different outcomes. In the coin-urn example, the PDF tells us that any probability 'p' for getting a head is just as likely as any other probability, hence the PDF is a constant value. When we integrate a constant PDF over a given range, such as \[0,1\], this operation helps calculate the overall probability for a certain event occurring, like getting a head on the first coin flip, which turned out to be \(1/2\).

Random Variables

Random variables are a core concept in probability theory. They aren't your typical variables from algebra that neatly solve for 'x'. Instead, think of them as placeholders for the outcomes of an experiment that involve chance. In the urn and coin scenario, the random variable 'p' represents the probability of flipping heads with any given coin.

The beauty of random variables lies in their ability to encapsulate uncertainty in a numerical form, which can then be manipulated mathematically to find probabilities. Random variables can be discrete, taking on specific values, or continuous, like 'p' in our case, which can take any value between 0 and 1. Understanding how to work with random variables gives us the superpower to handle probabilities in diverse scenarios, from flipping coins to predicting weather or stock market trends.