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: \(\frac{1}{3}\)