Question

There is a 55% probability that Ella is happy tomorrow if she's happy today, and there is a 90% probability that she's sad tomorrow if she's sad today. We know that seven days ago, Ella was sad. What is the probability that Ella was happy anytime in the last seven days?

Answer 1

P(sad taday | sad yesterday) = 0.90

P(happy today | sad yesterday) = 1-0.90 = 0.1

in a seven day period given she was sad on day before the period

P(haappy anytime) = P(happy on day 1) + P(happy on day 2 and sad all days before)+ P(happy on day 3 and sad all days before)+ P(happy on day 4 and sad all days before)+ P(happy on day 5 and sad all days before)+ P(happy on day 6 and sad all days before)+ P(happy on day 7 and sad all days before)

= 0.1 + 0.9*0.1 + 0.9*0.9*0.1 + 0.9*0.9*0.9*0.1 + 0.9*0.9*0.9*0.9*0.1 + 0.9*0.9*0.9*0.9*0.9*0.1 + 0.9*0.9*0.9*0.9*0.9*0.9*0.1

P(haappy anytime) = 0.5217

P.S. (please upvote)