Consider an um containing a large number of coins and...

Consider an um 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 um is such that if a coin is selected at random from the urn, then its $p$-value 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 is a head; (b) both flips are heads.

Consider an um 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 um is such that if a coin is selected at random from the urn, then its $p$-value 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 is a head;
(b) both flips are heads.

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Key Concepts

Uniform Distribution

A uniform distribution over an interval (in this case, [0,1]) means that every outcome within that range has an equal probability of occurring. When applied to the coin’s probability of landing heads, it implies that the bias or weight of any particular coin is equally likely to be any value between 0 and 1.

Law of Total Expectation

The law of total expectation is a fundamental tool for computing expected values when there is an underlying random parameter. In this problem, it is applied by calculating the overall probability of heads by averaging the probability p over its entire distribution, effectively integrating p with respect to its uniform density.

Mixture Distribution

A mixture distribution occurs when a random process involves parameters that are themselves random variables. Here, the coin flipping process is modeled as a mixture: first, a coin is selected with a random probability p (drawn from a uniform distribution), then that coin is flipped. The overall probabilities of outcomes are obtained by integrating over the distribution of p.

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