Question

Suppose that coin 1 has probability 0.8 of coming up heads, and coin 2 has probability 0.6 of coming up heads. If the coin flipped today comes up heads, then with equal probability we select coin 1 or coin 2 to flip tomorrow, and if it comes up tails, then we select coin 2 to flip tomorrow.

(a) If the coin initially flipped is coin 1, what is the probability that the coin flipped on the second day after the initial flip is coin 2?

(b) What proportion of flips use coin 1 and what proportion use 2 in the long run?

Answer 1

a.)If the coin initially flipped is coin 1, what is the probability that the coin flipped on the second day after the initial flip is coin 2

If initially flipped coin is coin 1 then coin1 has 0.8 prob of having Head and if it is head then on the very next day the 2nd coin to be flipped has prob of 0.5

while if the coin 1 is flipped and the 20% prob of tail comes up then the selected coin is 2 for the next flip

So the required probability = 0.8*0.5+0.2*1 = 0.6

b) In the long run the coin1count

0.8+0.8*0.5+0.8*0.5*0.5 +---- = 0.8/(1-0.5) = 1.6

0.2*0.5+0.2*0.5*0.5 + 0.2*0.5*0.5*0.5+--- = 0.01/(1-0.5) = 0.02

Total1=1.62

In the long run the coin2 count

0.6+0.6*0.5+0.6*0.5*0.5 +---- = 0.6/(1-0.5) = 1.2

0.2*0.5+0.2*0.5*0.5 + 0.2*0.5*0.5*0.5+--- = 0.01/(1-0.5) = 0.02

Total 2 =1.22

So the proportion of Coin 1= (1.62)/(1.62+1.22)=57.04%

So the proportion of Coin 2= (1.22)/(1.62+1.22)=42.96%

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