Suppose that coin 1 has probability 0.7 of coming up heads, and coin 2 has proba

Question

Suppose that coin 1 has probability 0.7 of coming up heads, and coin 2 has probability .6 of coming
up heads. If the coin ?ipped today comes up heads, then we select coin 1 to ?ip tomorrow, and
if it comes up tails, then we select coin 2 to ?ip tomorrow.

1. Find the transition matrix of the experiment.

2. If the coin initially ?ipped is equally likely to be coin 1 or coin 2, then what is the probability
that the coin ?ipped on the ?fth day after the initial ?ip is coin 1?

3. Suppose that the coin ?ipped on Monday comes up heads. What is the probability that
the coin ?ipped on Saturday of the same week also comes up heads?

4. Suppose that the coin ?ipped on Monday comes up heads, what is the expected number of
heads in ?ve days (after the ?ip on Friday of the same week)?

Explanation / Answer

Let the state be the number of coin flipped in that day: 1 for coin 1, 2 for coin 2. Then the transition probability matrix is P = .6 .4 .7 .3 2 1 1 2 and so, = .66 .34 .67 .33 P2 and = .666 .334 .667 .333 P3 Hence, the probability that a coin 1 is flipped on third day is [ ] 0.6665 2 1 3 21 3 11 P + P =

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