With explanation please Suppose that in two successive rounds of play in the gam

Question

With explanation please

Suppose that in two successive rounds of play in the game Candy Crush, scoring works as follows: Your score on the first round is uniformly distributed on the interval (0, 1); if your score on the first round is greater that 0.1, you are successful on the first round. If you are successful on the first round, your score on the second round is uniformly distributed on the interval (0.1, 1); otherwise, your score on the second round is again uniformly distributed on the interval (0, 1); in either case, you are successful on the second round if your score is greater than 0.2. Find the probability that you are successful on round 2.

Explanation / Answer

P(Successful in round 1)

= P(X > 0.1)

= 1 - P(X < 0.1)

= 1 - (0.1 - 0)/(1 - 0)

= 0.9

So,

P(Not successful in round 1) = 1 - 0.9 = 0.1

P(Successful in round 2)

= P(Successful in round 2 | successful in round 1) * P(Successful in round 1) + P(Successful in round 2 | not successful in round 1) * P(Not successful in round 1)

Now,

P(Successful in round 2 | successful in round 1)

= P(X > 0.2 | X > 0.1 in round 1)

= 1 - P(X < 0.2 | X > 0.1 in round 1)

= 1 - (0.2 - 0.1)/(1 - 0.1)

= 1 - 1/9

= 8/9

= 0.8889 (Approx)

And

P(Successful in round 2 | not successful in round 1)

= P(X > 0.2 | X < 0.1 in round 1)

= 1 - P(X < 0.2 | X < 0.1 in round 1)

= 1 - (0.2 - 0)/(1 - 0)

= 1 - 0.2

= 0.8

Hence,

P(Successful in round 2)

= (0.9)*(8/9) + (0.1)*(0.8)

= 0.8 + 0.08

= 0.88

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