More Than Two Outcomes

In Inception, Dom Cobb (i.e., Leonardo DiCaprio) uses a spinning top to determine if he is in a dream or reality. However, there is another layer: sometimes he is in a dream within a dream.

Suppose that Cobb is in reality 30% of the time, in a normal dream 50% of the time, and in a dream within a dream 20% of the time.

• If Cobb is in reality, the top stops spinning 90% of the time and keeps spinning 10% of the time.
• If Cobb is in a normal dream, the top keeps spinning 80% of the time and stops 20% of the time.
• If Cobb is in a dream within a dream, the top keeps spinning 95% of the time and stops 5% of the time.

Question:

• (2 points) Given that Cobb observes the top still spinning, what is the probability that he is in a dream within a dream?

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